The inverse of the matrix \(\begin{bmatrix} 2 & 5 & 0 \\ 0 & 1 & 1 \\ -1 & 0 &3 \end{bmatrix} \)
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\(\begin{bmatrix} 3 & -15 &5 \\ -1 &6 &-2 \\ 1 &-5 &2 \end{bmatrix} \)
\(\begin{bmatrix} 3 & -1 &1 \\ -15 &6 &-5 \\ 5 &-2 &2 \end{bmatrix} \)
\(\begin{bmatrix} 3 & -15 &5 \\ -1 &6 &-2 \\ 1 &-5 &-2 \end{bmatrix} \)
\(\begin{bmatrix} 3 & -5 &5 \\ -1 &-6 &-2 \\ 1 &-5 &2 \end{bmatrix}\)
If P and Q are symmetric matrices of the same order then PQ-QP is
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Zero matrix
Identity matrix
Skew symmetric matrix
Symmetric matrix
If \(3A+4B'=\begin{bmatrix} 7 &-10 &17 \\ 0 & 6 &31 \end{bmatrix} \) and \(2B-3A'=\begin{bmatrix} -1 & 18\\ 4& 0\\ -5&-7 \end{bmatrix} \) then B =
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(A) \(\begin{bmatrix} -1 & -18\\ 4& -16\\ -5&-7 \end{bmatrix} \)
\(\begin{bmatrix} 1 & 3\\ -1& 1\\ 2&4 \end{bmatrix} \)
\(\begin{bmatrix} 1 & 3\\ -1& 1\\ 2&-4 \end{bmatrix} \)
\(\begin{bmatrix} 1 & -3\\ -1& 1\\ 2&4 \end{bmatrix}\)
If \(A=\begin{bmatrix} 1 &3 \\ 4& 2 \end{bmatrix},B=\begin{bmatrix} 2 &-1 \\ 1 & 2 \end{bmatrix} \) , Then \(\left | ABB \,' \right |=\)
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100
50
250
-250
If the value of a third order determinant is 16, then the value of the determinant formed by replacing each of its elements by its cofactor is
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256
96
16
48
\(\large\int x^3\,sin\,3xdx= \)
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\(\Large-\frac{x^3.cos\,3x}{3}+\frac{x^2.sin\,3x}{3}+\frac{2xcos\,3x}{9}-\frac{2sin\,3x}{27}+C \)
\(\Large-\frac{x^3.cos\,3x}{3}-\frac{x^2.sin\,3x}{3}+\frac{2xcos\,3x}{9}-\frac{2sin\,3x}{27}+C \)
\(\Large-\frac{x^3.cos\,3x}{3}+\frac{x^2.sin\,3x}{3}-\frac{2xcos\,3x}{9}-\frac{2sin\,3x}{27}+C \)
\(\Large\frac{x^3.cos\,3x}{3}+\frac{x^2.sin\,3x}{3}-\frac{2xcos\,3x}{9}-\frac{2sin\,3x}{27}+C\)
The area of the region above X-axis included between the parabola \(y^2 = x\) and the circle \( x^2 + y^2 = 2x\) in square units is
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\(\Large\frac{2}{3}-\frac{\pi}{4} \)
\(\Large\frac{\pi}{4}-\frac{3}{2} \)
\(\Large\frac{\pi}{4}-\frac{2}{3} \)
\(\Large\frac{3}{2}-\frac{\pi}{4}\)
The area of the region bounded by \(Y-axis, y = cos x\) and \(y = sin x\) , \(\large0\leq x\leq \frac{\pi}{2}\) is
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\(\sqrt2 +1\,\, Sq. units\)
\(\sqrt2 -1\,\, Sq. units\)
\(2-\sqrt2 \,\, Sq. units\)
\(\sqrt2 \,\, Sq. units\)
The integrating factor of the differential equation \((2x + 3y^2) \,dy = y \,dx \,\,(y > 0)\) is
Please choose your answer from the right side options
\( \Large{\frac{1}{x}}\)
\(\Large\frac{1}{e^y}\)
\(\Large\frac{1}{y^2}\)
\(-\Large\frac{1}{y^2}\)
If \(2^x+2^y=2^{x+y}\) , then \(\Large\frac{dy}{dx}\) is
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\(2^{y-x}\)
\(-2^{y-x}\)
\(2^{x-y}\)
\(\Large\frac{2^y-1}{2^x-1}\)
If \(f(x)=sin^{-1}\left ({ \Large\frac{2x}{1+x^2}} \right ) \) , then \(f'(\sqrt{3}) \) is
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\(-\Large\frac{1}{2} \)
\(\Large\frac{1}{2} \)
\(\Large\frac{1}{\sqrt{3}} \)
\(-\Large\frac{1}{\sqrt{3}}\)
The right hand and left hand limit of the function are respectively
\(f{(x)}=\left\{\begin{matrix} \Large\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1} &,\,if\,x\neq 0 \\ 0, & if\,x= 0 \end{matrix}\right.\)
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1 and 1
1 and -1
-1 and -1
-1 and 1
If \(y=2x^{n+1}+{\Large\frac{3}{x^n}} \) , then \(x^2{\Large\frac{d^2y}{dx^2}} \) is
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6n(n+1)y
n(n+1)y
\(x{\Large\frac{dy}{dx}}+y\)
y
If the curves \(2x=y^2\) and 2xy=K intersect perpendicularly, then the value of \(K^2\) is
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4
\(2\sqrt2\)
2
8
If \(\left ( xe \right )^y=e^x \) , then is \(\Large\frac{dy}{dx} \) is =
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\(\Large\frac{log\,x}{(1+log\,x)^2} \)
\(\Large\frac{1}{(1+log\,x)^2} \)
\(\Large\frac{log\,x}{(1+log\,x)} \)
\(\Large\frac{e^x}{x(y-1)}\)
If the side of a cube is increased by 5%, then the surface area of a cube is increased by
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10%
60%
6%
20%
The value of \(\int {\large\frac{1+x^4}{1+x^6}}dx \) is
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\(tan^{-1}\,x+tan^{-1}\,x^3+C \)
\(tan^{-1}\,x+{\Large\frac{1}{3}}tan^{-1}\,x^3+C \)
\(tan^{-1}\,x-{\Large\frac{1}{3}}tan^{-1}\,x^3+C \)
\(tan^{-1}\,x+{\Large\frac{1}{3}}tan^{-1}\,x^2+C\)
The maximum value of \(\Large\frac{log_ex}{x}\) , if x > 0 is
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e
1
\(\Large\frac{1}{e}\)
\(-\Large\frac{1}{e}\)
The value of \(\frac{2(cos\,75^{\circ}+isin\,75^{\circ})}{0.2(cos\,30^{\circ}+isin\,30^{\circ})}\) is
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\(\frac{5}{\sqrt{2}}(1+i)\)
\(\frac{10}{\sqrt{2}}(1+i)\)
\(\frac{10}{\sqrt{2}}(1-i)\)
\(\frac{5}{\sqrt{2}}(1-i)\)
\(\frac{1}{\sqrt{2}}(1+i)\)
If the conjugate of a complex number \(z\) is \(\frac{1}{i-1}\) , then \(z\) is
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\(\frac{1}{i-1}\)
\(\frac{1}{i+1}\)
\(\frac{-1}{i-1}\)
\(\frac{-1}{i+1}\)
\(\frac{1}{i}\)
The value of \(\left ( i^{18}+\left ( \frac{1}{i} \right )^{25}\right )^{3}\) is equal to
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\(\frac{1+i}{2}\)
\(2+2i\)
\(\frac{1-i}{2}\)
\(\sqrt2-\sqrt2 i\)
\(2-2i\)
The modulus of \(\frac{1+i}{1-i}-\frac{1-i}{1+i}\) is
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\(2\)
\(\sqrt2\)
\(4\)
\(8\)
\(10\)
If \(z=e^\frac{i4\pi }{3}\) , then \((z^{192}+z^{194})^3 \) is equal to
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\(-2\)
\(-1\)
\(-i\)
\(-2i\)
\(0\)
If \(a\) and \(b\) are real numbers and \((a+ib)^{11}=1+3i\) , then \((b+ia)^{11}\) is equal to
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\(i+3\)
\(1+3i\)
\(1-3i\)
\(0\)
\(-i-3\)
If \(\alpha \neq \beta ,\alpha ^2=5\alpha -3,\beta ^2=5\beta -3\) , then the equation having \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) as its roots is
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\(3x^2-19x-3=0\)
\(3x^2+19x-3=0\)
\(x^2+19x+3=0\)
\(3x^2-19x-19=0\)
\(3x^2-19x+3=0\)
The focus of the parabola \(y^2-4y-x+3=0 \) is
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\(\left ( \frac{3}{4},2 \right )\)
\(\left ( \frac{3}{4},-2 \right )\)
\(\left ( 2, \frac{3}{4} \right )\)
\(\left ( \frac{-3}{4},2 \right )\)
\(\left ( 2,\frac{-3}{4} \right )\)
If \(f:R\rightarrow (0,\infty )\) is an increasing function and if \(lim_{x\rightarrow 2018}\frac{f(3x)}{f(x)}=1\) , then \(lim_{x\rightarrow 2018}\frac{f(2x)}{f(x)}\) is equal to
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\(\frac{2}{3}\)
\(\frac{3}{2}\)
\(2\)
\(3\)
\(1\)
The equation of the curve passing through the point (1, 1) such that the slope of the tangent at any point (x, y) is equal to the product of its co-ordinates is
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\(2\,log\,y=x^2-1 \)
\(2\,log\,x=y^2-1 \)
\(2\,log\,x=y^2+1 \)
\(2\,log\,y=x^2+1\)
Foot of the perpendicular drawn from the point (1, 3, 4) to the plane \(2x - y + z + 3 = 0\) is
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(1, 2, - 3)
(-1, 4, 3)
\( (-3, 5, 2) \)
\((0, -4, -7)\)
Acute angel between the line \(\Large\frac{x-5}{2}=\frac{y+4}{-1}=\frac{z+4}{1} \) and the plane \(3x - 4y - z + 5 = 0 \) is
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\(\Large\cos^{-1}\left ( \frac{5}{2\sqrt{13}} \right ) \)
\(\Large\cos^{-1}\left ( \frac{9}{\sqrt{364}} \right ) \)
\(\Large\sin^{-1}\left ( \frac{5}{2\sqrt{13}} \right ) \)
\(\Large\sin^{-1}\left ( \frac{9}{\sqrt{364}} \right )\)
Non of the Above
The distance of the point (1, 2, 1) from the line \(\Large\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{2} \) is
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\(\Large\frac{\sqrt{5}}{3} \)
\(\Large\frac{2\sqrt{3}}{5} \)
\(\Large\frac{{20}}{3} \)
\(\Large\frac{2\sqrt{5}}{3}\)
XY – plane divides the line joining the points A (2, 3, - 5) and B (-1, -2, -3) in the ratio
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5 : 3 internally
2 : 1 internally
5 : 3 externally
3 : 2 externally
The shaded region in the figure is the solution set of the inequations.
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\(\large4x + 5y ≤ 20, 3x + 10y ≤ 30, x ≤ 6, x, y ≥ 0 \)
\(\large4x + 5y ≥ 20, 3x + 10y ≤ 30, x ≤ 6, x, y ≥ 0 \)
\(\large4x + 5y ≤ 20, 3x + 10y ≤ 30, x ≥ 6, x, y ≥ 0 \)
\(\large4x + 5y ≥ 20, 3x + 10y ≤ 30, x ≥ 6, x, y ≥ 0\)
The constant term in the expansion of \(\begin{bmatrix} 3x+1 &2x-1 &x+2 \\ 5x-1&3x+2 & x+1\\ 7x-2 & 3x+1 & 4x-1 \end{bmatrix}\) is
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– 10
0
6
2
If \([x]\) represents the greatest integer function and \(f (x) = x - [x] – cos x\) then \(\Large f'(\Large\frac{\pi}{2})\)=
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2
0
does not exist
1
If \(\Large f(x)=\left\{\begin{matrix} \frac{sin\,3x}{e^{2x}-1} &;x\neq 0 \\ k-2 &; x=0 \end{matrix}\right.\) is Continuous at x = 0, then k =
Please choose your answer from the right side options
\(\Large\frac{1}{2}\)
\(\Large\frac{3}{2}\)
\(\Large\frac{2}{3}\)
\(\Large\frac{9}{5}\)
Non of the Above
If \(\Large f(x)=sin^{-1}\left [ \frac{2^{x+1}}{1+4^x} \right ]\) , then \(f’ (0) =\)
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\(\Large \frac{2\,log\,2}{5} \)
\(\Large2\,log\,2 \)
\(\Large \frac{4\,log\,2}{5} \)
\(\Large log\,2 \)
If \(x=1\,sec^2\,\theta,y=a\,tan^2\theta \) then \(\Large \frac{d^2y}{dx^2}=\)
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0
2a
4
1
If \(α\) and \(β\) are roots of the equation \(\large x^2 = x + 1 = 0\) then \(α^ 2 + β ^ 2\) is
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\(\Large\frac{-1-i\sqrt{3}}{2} \)
\(1 \)
\(-1 \)
\(\Large\frac{-1+i\sqrt{3}}{2}\)
The number of 4 digit numbers without repetition that can be formed using the digits 1, 2, 3, 4, 5, 6, 7 in which each number has two odd digits and two even digits is
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450
432
454
436
The number of terms in the expansion of \(\large(x^2+y^2)^{25}-(x^2-y^2)^{25}\) after simplification is
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26
0
50
13
The third term of a G.P. is 9. The product of its first five terms is
Please choose your answer from the right side options
\(3^{10}\)
\(3^5\)
\(3^{12}\)
\(3^9\)
A line cuts off equal intercepts on the co-ordinate axes. The angle made by this line with the positive direction of X-axis is
Please choose your answer from the right side options
120°
45°
135°
90°
The eccentricity of the ellipse \(9x^2 + 25y^2 = 225\) is
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\(\Large\frac{3}{4} \)
\(\Large\frac{4}{5} \)
\(\Large\frac{9}{16} \)
\(\Large\frac{3}{5}\)
\(\Large\sum_{r=1}^{n}(2r-1)=x \) then \(\Large\lim_{n\rightarrow \infty }\left [\frac{1^3}{x^2}+\frac{2^3}{x^2}+\frac{3^3}{x^2}+......+\frac{n^3}{x^2} \right ]=\)
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1
\(\Large\frac{1}{2}\)
4
\(\Large\frac{1}{4}\)
The negative of the statement “All continuous functions are differentiable.”
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Some continuous functions are not differentiable
All continuous functions are not differentiable.
All differentiable functions are continuous.
Some continuous functions are differentiable
Mean and standard deviation of 100 items are 50 and 4 respectively. The sum of all squares of the items is
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266000
251600
261600
256100
Two letters are chosen from the letters of the word ‘EQUATIONS’. The probability that one is vowel and the other is consonant is
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\(\Large\frac{3}{9}\)
\(\Large\frac{8}{9}\)
\(\Large\frac{5}{9}\)
\(\Large\frac{4}{9}\)
\(f: R → R\) and \(g : [0, ∞) → R\) is defined by \(f(x) = x^2\) and \(g(x) =\sqrt{x}\) . Which of the following is not true?
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fog (2) = 2
gof (4) = 4
gof (-2) = 2
fog (-4) = 4
If \(A = \{x| x ∈ N, x ≤ 5\}, B =\{ x | x ∈ Z, x^2 – 5x + 6 = 0\}\), then the number of onto functions from A to B is
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30
2
32
23
On the set of positive rationals, a binary operation \(\large *\) is defined by \(\Large a * b =\frac{2ab}{5}\). If \(\large2 * x = 3^{-1}\) then \(x=\)
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\(\Large\frac{2}{5} \)
\(\Large\frac{1}{6} \)
\(\Large\frac{125}{48} \)
\( \Large\frac{5}{12}\)
\(\Large cos\left [ 2\,sin^{-1}\frac{3}{4}+ cos^{-1}\frac{3}{4}\right ]= \)
Please choose your answer from the right side options
\(\Large\frac{3}{5} \)
\(-\Large\frac{3}{4} \)
does not exist
\(\Large\frac{3}{4}\)
If \(\large a+\Large\frac{\pi}{2}<2\,tan^{-1}\,x+3\,cot^{-1}\,x<b\) then ‘a’ and ‘b’ are respectively.
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0 and 2π
0 and π
\(-\Large \frac{\pi}{2}\) and \(\Large \frac{\pi}{2}\)
\(\Large \frac{\pi}{2}\)and \(\Large\2pi\)
If \(|3x – 5| ≤ 2\) then
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\(\Large1\leq x\leq \frac{9}{3} \)
\(\Large-1\leq x\leq \frac{7}{3}\)
\(\Large-1\leq x\leq \frac{9}{3} \)
\(\Large1\leq x\leq \frac{7}{3}\)
A random variable ‘X’ has the following probability distribution:
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(X) | k-1 | 3k | k | 3k | \(3k^2\) | \(k^2\) | \(k^2+k\) |
Then the value of k is
Please choose your answer from the right side options
\(\Large \frac{2}{7}\)
\(\Large\frac{1}{5}\)
\(\Large\frac{1}{10}\)
\(-2\)
If A and B are two events of a sample space S such that P(A) = 2.0, P(B) = 0.6 and P(A|B) = 0.5 then P(A’|B)=
Please choose your answer from the right side options
\(\Large\frac{1}{2}\)
\(\Large\frac{3}{10}\)
\(\Large\frac{1}{3}\)
\(\Large\frac{2}{3}\)
If ‘X’ has a binomial distribution with parameters n = 6, p and P(X = 2) = 12, P(X = 3) = 5 then P =
Please choose your answer from the right side options
\(\Large\frac{1}{2}\)
\(\Large\frac{5}{12}\)
\(\Large\frac{5}{16}\)
\(\Large\frac{16}{21}\)
Non of the Above
A man speaks truth 2 out of 3 times. He picks one of the natural numbers in the set S = {1, 2, 3, 4, 5, 6, 7} and reports that it is even. The probability that it is actually even is
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\(\Large\frac{1}{10}\)
\(\Large\frac{2}{5}\)
\(\Large\frac{3}{5}\)
\(\Large\frac{1}{5}\)
The order of the differential equation \(\large y=C_1e^{C_2+x}+C_3e^{C_4+x}\) is
Please choose your answer from the right side options
3
1
4
2
If \(\large\left | \vec{a} \right |=16,\left | \vec{b} \right |=4 \) , then, \(\large \sqrt{\left | \vec{a}\times \vec{b} \right |^2+ \left | \vec{a}. \vec{b} \right |^2}=\)
Please choose your answer from the right side options
16
4
64
8
If the angle between \(\vec{a} \)and \(\vec{b} \) is \(\Large\frac{2\pi}{3} \) and the projection of \(\vec{a} \) in the direction of \(\vec{b} \) is -2, the \(\left | \vec{a} \right |\)=
Please choose your answer from the right side options
2
4
1
3
A unit vector perpendicular to the plane containing the vectors \(\hat{i}+2\hat{j}+\hat{k} \) and \(-2\hat{i}+\hat{j}+3\hat{k} \) is
Please choose your answer from the right side options
\(\Large\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}} \)
\(\Large\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}} \)
\(\Large\frac{-\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}} \)
\(\Large\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}\)
\(\large\left [ \vec{a}+2\vec{b}-\vec{c} ,\vec{a}-\vec{b},\vec{a}-\vec{b}-\vec{c}\right ]= \)
Please choose your answer from the right side options
\(\large2\left [ \vec{a},\vec{b},\vec{c} \right ] \)
0
\(\large3\left [ \vec{a},\vec{b},\vec{c} \right ] \)
\(\large\left [ \vec{a},\vec{b},\vec{c} \right ]\)
\(\Large\sqrt[3]{y}\sqrt{x}\,\,\,\sqrt[6]{(x+y)^5} \), then \(\Large\frac{dy}{dx}=\)
Please choose your answer from the right side options
\(\large x-y\)
\(\Large\frac{x}{y}\)
\(\Large\frac{y}{x}\)
\(\Large x+y\)
Rolle’s theorem is not applicable in which one of the following cases?
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\( f (x) = |x| \) in \( [-2, 2]\)
\(f(x) = x^2 – 4x + 5\) in [1, 3]
\(f (x) = [x]\) in [2.5, 2.7]
\(f(x) = x^2 – x\) in [0,1]
The interval in which the function \(f (x) = x^3 – 6x^2 + 9x + 10\) is increasing in
Please choose your answer from the right side options
[1, 3]
\((-∞, 1) ∪ (3, ∞)\)
\((-∞, -1] ∪ [3, ∞)\)
\((-∞, 1] ∪ [3, ∞)\)
The sides of an equilateral triangle are increasing at the rate of 4 cm/sec. The rate at which its area is increasing, when the side is 14 cm.
Please choose your answer from the right side options
\(42\, cm^2 /sec\)
\(10\sqrt{3}\, cm^2 /sec\)
\(14 \,cm^2 /sec\)
\(14\sqrt{3}\, cm^2 /sec\)
Non of the Above
The value of \(\sqrt{24.99}\) is
Please choose your answer from the right side options
5.001
4.999
4.897
4.899
Non of the Above
\(\Large\int_{-3}^{3} cot^{-1}x\,dx =\)
Please choose your answer from the right side options
\(\large6 \pi\)
\(\large3\pi\)
3
0
\(\Large\int \frac{1}{\sqrt{x}+x\sqrt{x}}\,dx= \)
Please choose your answer from the right side options
\( tan^{-1}\sqrt{x} + C \)
\(2\, log ( \sqrt{x }+ 1)+ C \)
\(2 tan^{-1}\sqrt{x} + C \)
\({\Large\frac{1}{2}}tan^{-1}\sqrt{x}+C \)
\({\Large\int \frac{2x-1}{(x-1)(x^2)(x-3)}}dx=A\,log\,\left | x-1 \right |+B\,log\,\left | x+2 \right |+C\,log\,\left | x-3 \right |+K\)
Then A, B, C are respectively.
Please choose your answer from the right side options
\(\Large\frac{1}{6},\frac{-1}{3},\frac{1}{3} \)
\(\Large\frac{-1}{6},\frac{1}{3},\frac{1}{3} \)
\(\Large\frac{-1}{6},\frac{-1}{3},\frac{1}{2} \)
\(\Large\frac{1}{6},\frac{1}{3},\frac{1}{5} \)
\(\Large\int_{0}^{2}\left [ x^2 \right ]dx= \)
Please choose your answer from the right side options
\(\large5-\sqrt{2}+\sqrt{3} \)
\(\large5-\sqrt{2}-\sqrt{3} \)
\(-\large5-\sqrt{2}-\sqrt{3}\)
\( \large5+\sqrt{2}-\sqrt{3}\)
\(\Large\int_{0}^{1}\sqrt{\frac{1+x}{1-x}}dx=\)
Please choose your answer from the right side options
\(\Large\frac{\pi}{2}\)
\({\Large\frac{\pi}{2}}-1\)
\(\Large\frac{1}{2}\)
\({\Large\frac{\pi}{2}}+1\)
If U is the universal set with 100 element; A and B are two set such that n(A) = 50, n (B) = 60, n (A∩B) = 20 then n (A’∩B’) =
Please choose your answer from the right side options
90
40
10
20
The domain of the function \(f : R → R\) defined by \(f (x)=\sqrt{x^2-7x+12}\) is
Please choose your answer from the right side options
\((-∞, 3] ∩ [4, ∞)\)
\((-∞, 3] ∪ [4, ∞)\)
(3, 4)
\((-∞, 3] ∪ (4, ∞)\)
If \(cos \,x = |sin \,x|\) then, the general solution is
Please choose your answer from the right side options
\(\large x=n\pi+(-1)^n\,\,{\Large\frac{\pi}{4}},n\epsilon Z \)
\(\large x=n\pi\pm \,\,{\Large\frac{\pi}{4}},n\epsilon Z \)
\(\large x= (2n=1)\pi\pm \,\,{\Large\frac{\pi}{4}},n\epsilon Z \)
\(\large x=2n\pi\pm \,\,{\Large\frac{\pi}{4}},n\epsilon Z\)
\(\large \sqrt{3}\,\,cosec\,20^{\circ}-sec\,20^{\circ}=\)
Please choose your answer from the right side options
4
2
1
3
If \(P (n) : 2^n < n!\) , Then the smallest positive integer for which P (n) is true, is
Please choose your answer from the right side options
4
2
5
3
The value of \(\int e^{sin\,x}\,sin\,2x\,dx \) is
Please choose your answer from the right side options
\(2e^{sin\,x}(sin\,x-1)+C \)
\(2e^{sin\,x}(sin\,x+1)+C \)
\(2e^{sin\,x}(cos\,x+1)+C \)
\(2e^{sin\,x}(cos\,x-1)+C\)
The value of \(\int_{-\frac{1}{2}}^{\frac{1}{2}}cos^{-1}\,x\,dx \) is
Please choose your answer from the right side options
π
\(\Large\frac{\pi}{2} \)
1
\(\Large\frac{\pi^2}{2}\)
If \(\int{ \Large\frac{3x+1}{(x-1)(x-2)(x-3)}}\,dx=\,A\,log\,\left | x-1 \right |B\,log\,\left | x-2 \right |+C\,log\,\left | x-3 \right |\) , then the values of A, B and C are respectively
Please choose your answer from the right side options
5, -7, -5
2, -7, -5
5, -7, 5
2, -7, 5
The value of \({\Large\int_{0}^{1}}{\Large\frac{log(1+x)}{1+x^2}}dx \) is
Please choose your answer from the right side options
\({\Large\frac{\pi}{2}}log\,2 \)
\({\Large\frac{\pi}{4}}log\,2 \)
\(\Large\frac{1}{2} \)
\({\Large\frac{\pi}{8}}log\,2\)
The area of the region bounded by the curve \(y^2 =8x\) and the line \(y=2x\) is
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\(\Large\frac{16 }{ 3}\) sq .units
\(\Large\frac{4}{ 3}\) sq .units
\(\Large\frac{3 }{ 4}\) sq. units
\(\Large\frac{8 }{ 3}\) sq .units
The value of \({\Large\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{cos\,x}{1+e^x}}dx\) is
Please choose your answer from the right side options
2
0
1
-2
The order of the differential equation obtained by eliminating arbitrary constants in the family of curves \(c_1y=(c_2+c_3)e^{x+c_4}\) is
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1
2
3
4
The general solution of the differential equation \(\large x^2dy-2xydx=x^4\,cos\,xdx \) is
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\(\large y=x^2\,sin\,x+cx^2 \)
\(\large y=x^2\,sin\,x+c \)
\(\large y=\,sin\,x+cx^2 \)
\(\large y=\,cos\,x+cx^2\)
The area of the region bounded by the line \(y=2x+1\), x−axis and the ordinates x=−1 and x=1 is
Please choose your answer from the right side options
\(\Large\frac{9}{4}\)
2
\(\Large\frac{5}{2}\)
5
The two vectors \(\hat{i}+\hat{j}+\hat{k} \) and \(\hat{i}+3\hat{j}+5\hat{k} \) represent the two sides \(\overline{AB} \) and \(\overline{AC} \) respectively of a \(ΔABC\). The length of the median through A is
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\(\Large\frac{\sqrt{14}}{2} \)
14
7
\(\sqrt{14}\)
If \(\vec{a} \) and \(\vec{b} \) are unit vectors and \(θ\) is the angle between \(\vec{a} \) and \(\vec{b} \) then \(sin\,\Large\frac{\theta}{2}\)
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\(\left | \vec{a} +\vec{b} \right | \)
\(\Large\frac{\left | \vec{a} +\vec{b} \right | }{2} \)
\(\Large\frac{\left | \vec{a} -\vec{b} \right | }{2} \)
\(\left | \vec{a} -\vec{b} \right |\)
The curve passing through the point (1, 2) given that the slope of the tangent at any point (x,y) is \(\Large\frac{3x}{y}\) represents
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Circle
Parabola
Ellipse
Hyperbola
If \(\large\left | \vec{a}+\vec{b} \right |^2+\left | \vec{a}\cdot \vec{b} \right |^2=144\left | \vec{a} \right |=6 \) then \(\large\left | \vec{b} \right |\) is equal to
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6
3
2
4
The point (1, -3, 4) lies in the octant
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Second
Third
Fourth
Eighth
If the vectors \(2\hat{i}-3\hat{j}+4\hat{k} \,,2\hat{i}+\hat{j}-\hat{k} \) and \(\lambda \hat{i}-\hat{j}+2\hat{k}\) are coplanar, then the value of \(\lambda\) is
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6
-5
-6
5
The distance of the point (1, 2, -4) from the line \(\Large\frac{x-3}{2}=\frac{y-3}{3}=\frac{z+5}{6} \) is
Please choose your answer from the right side options
\(\Large\frac{293}{7} \)
\(\Large\frac{\sqrt{293}}{7} \)
\(\Large\frac{293}{49} \)
\(\Large\frac{\sqrt{293}}{49}\)
The sine of the angle between the straight line \(\Large\frac{x-2}{3}=\frac{3-y}{-4}=\frac{z-4}{5} \) and the plane \(2x−2y+z=5\) is
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\(\Large\frac{3}{\sqrt{30}} \)
\(\Large\frac{{3}}{50} \)
\(\Large\frac{4}{5\sqrt2} \)
\(\Large\frac{\sqrt{2}}{10}\)
If a line makes an angle of \(\Large\frac{\pi}{3}\) with each of x and and y−axis, then the acute angle made by z−axis is
Please choose your answer from the right side options
\(\Large \frac{\pi}{4} \)
\(\Large\frac{\pi}{6} \)
\(\Large\frac{\pi}{3} \)
\(\Large\frac{\pi}{2}\)
Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0). Let z = px = qy , where p, q>0. Condition on p and q so that the minimum of z occurs at (3, 0) and (1, 1) is
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p=2q
\(p=\Large\frac{q}{2}\)
p=3q
p=q
The feasible region of an LPP is shown in the figure. If \(Z = 11x + 7y\) , then the maximum value of Z occurs at
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(0,5)
(3,3)
(5,0)
(3,2)
A die is thrown 10 times, the probability that an odd number will come up at least one time is
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\(\Large\frac{1}{1024} \)
\(\Large\frac{1023}{1024} \)
\(\Large\frac{11}{1024} \)
\(\Large\frac{1013}{1024}\)
If A and B are two events such that \(P(A)={\Large\frac{1}{3}},P(B)=\Large\frac{1}{2} \) and \(P(A\cap B)=\Large\frac{1}{6} \) , then \(P(A'/B)\) is
Please choose your answer from the right side options
\(\Large\frac{2}{3} \)
\(\Large\frac{1}{3} \)
\(\Large\frac{1}{2} \)
\(\Large\frac{1}{12}\)
Events \(E_1\) and \(E_2\) from a partition of the sample space S. A is any event such that \(P(E_1)=P(E_2)={\Large\frac{1}{2}},P(E_2/A)={\Large\frac{1}{2}} \)and \(P(A/E_2)={\Large\frac{2}{3}} \) , then \(P(E_1/A) \) is
Please choose your answer from the right side options
\({\Large\frac{1}{2}} \)
\({\Large\frac{2}{3}} \)
\(1 \)
\({\Large\frac{1}{4}}\)
The probability of solving a problem by three persons A, B and C independently is \(\Large\frac{1}{2},\Large\frac{1}{4}\) and \(\Large\frac{1}{3}\) respectively. Then the probability of the problem is solved by any two of them is
Please choose your answer from the right side options
\(\Large\frac{1}{12}\)
\(\Large\frac{1}{4}\)
\(\Large\frac{1}{24}\)
\(\Large\frac{1}{8}\)
If n(A) = 2 and total number of possible relations from Set A to set B is 1024, then n(B) is
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512
20
10
5
The value of \(\large sin^2\,51^{\circ}+sin^2\,39^{\circ}\) is
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1
0
\(sin12^{\circ} \)
\(cos12^{\circ}\)
If \(tan A + cot A = 2\), then the value of \(tan^4 A + cot ^4A = \)
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2
1
4
5
If A = {1,2,3,4,5,6}, then the number of subsets of A which contain at least two elements is
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64
63
57
58
If z = x+iy, then the equation |z+1| = |z-1| represents
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a circle
a parabola
x-axis
y-axis
The value of \(^{16}C_9+^{16}C_{10}-^{16}C_6-^{16}C_7 \) is
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0
1
\(^{17}C_{10}\)
\(^{17}C_{3}\)
The number of terms in the expansion of \((x+y+z)^{10}\) is
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66
142
11
110
If \(P(n):2^n<n!\) Then the smallest positive integer for which P(n) is true , is
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2
3
4
5
The two lines \(lx+my=n\) and \(l'x+m'y=n'\) are perpendicular if
Please choose your answer from the right side options
\(ll'+mm'=0\)
'>\( lm'+ml'\)
'>\(lm+l'm'=0\)
'>\(lm'+ml'=0\)
If the parabola \(x^2=4ay\) passes through the point (2, 1), then the length of the latus rectum is
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1
4
2
8
If the sum of n terms of an A.P is given by \(S_n=n^2 +n\), then the common difference of the A.P is
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4
1
2
6
The negation of the statement "For all real numbers x and y, \(x + y = y + x\) " is
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For all real numbers x and y, \(x+y≠y+x\)
For some real numbers x and y, \(x+y = y+x\)
For some real number x and y, \(x + y ≠ y + x\)
for some real numbers x and y, \(x-y=y-x\)
The standard deviation of the data 6, 7, 8, 9, 10 is
Please choose your answer from the right side options
\(\sqrt2\)
\(\sqrt{10}\)
2
10
\(\lim_{x\rightarrow 0}\left ( \Large\frac{tan\,x}{\sqrt{2x+4}-2} \right )\) is equal to
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2
3
4
6
If a relation R on the set {1, 2, 3} be defined by R={(1, 1)}, then R is
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Reflexive and symmetric
Reflexive and transitive
symmetric and transitive
Only symmetric
Let \(f:[2,∞)→R\) be the function defined \(f(x)=x^2 −4x+5\) , then the range of f is
Please choose your answer from the right side options
\((−∞,∞)\)
\([1,∞) \)
\((1,∞)\)
\([5,∞)\)
If A, B, C are three mutually exclusive and exhaustive events of an experiment such that P(A)= 2 P(B) = 3P(C), then P(B) is equal to
Please choose your answer from the right side options
\(\Large\frac{1}{11} \)
\(\Large\frac{2}{11} \)
\(\Large\frac{3}{11} \)
\(\Large\frac{4}{11}\)
The domain of the function defined by \(f(x)=cos^{-1}\sqrt{x-1 }\)
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[1, 2]
[0, 2]
[-1, 1]
[0, 1]
The value of \(cos\left ( sin^{-1}{\Large\frac{\pi}{3}}+cos^{-1} {\Large\frac{\pi}{3}}\right )\) is
Please choose your answer from the right side options
0
1
-0
Does not exist
If \(A=\begin{pmatrix} 0 &0 &1 \\ 0 & 1 & 0\\ 1 &0 &0 \end{pmatrix}\) then \(A^4\) is equal to
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A
2A
I
4A
If A = {a,b,c}, then the number of binary operations on A is
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3
\(3^6 \)
\(3^3 \)
\(3^9\)
If \(\begin{pmatrix} 2 &1 \\ 3 & 2 \end{pmatrix}A=\begin{pmatrix} 1 &0 \\ 0&1 \end{pmatrix} \) , then the matrix a is
Please choose your answer from the right side options
\(\begin{pmatrix} 2 &1 \\ 3&2 \end{pmatrix} \)
\(\begin{pmatrix} 2 &-1 \\ -3&2 \end{pmatrix} \)
\(\begin{pmatrix} -2 &1 \\ 3&-2 \end{pmatrix} \)
\(\begin{pmatrix} 2 &-1 \\ 3&2 \end{pmatrix}\)
If \(f(x)=\begin{vmatrix} x^3-x &a+x & b+x\\ x-a& x^2-x & c+x\\ x-b& x-c &0 \end{vmatrix}\) then
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f(1)=0
f(2)=0
f(0)=0
f(-1)=0
If A and B are square matrices of same order and B is a skew symmetric matrix, then \(A’BA\) is
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Symmetric matrix
Null matrix
Diagonal matrix
Skew symmetric matrix
If A is a square matrix of order 3 and |A| = 5, then |A adj.A| is
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5
125
25
625
If \(f(n)=\left\{\begin{matrix} {\Large\frac{1-cos\,Kx}{x\,sinx}} ,&if\,x\neq 0 \\ {\Large\frac{1}{2}} &, if\,x=0 \end{matrix}\right.\)
Please choose your answer from the right side options
\(\pm \Large\frac{1}{2}\)
0
\(±2\)
\(±1\)
If \(a_1a_2a_3......a_9 \) are in A.P. then the value of \(\begin{vmatrix} a_1 & a_2 &a_3 \\ a_4 &a_5 & a_6\\ a_7 & a_8 & a_9 \end{vmatrix} \) is
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\({\Large\frac{9}{2}}(a_1+a_0) \)
\((a_1+a_9) \)
\(log_e(log_ee)\)
1
\(\Large f(x) = \frac{x}{e^x -1} + \frac{x}{2} + 2 \ cos^3 \frac{x}{2}\) on R - {0} is
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One one function
Bijection
Algebraic function
Even function
Consider the following lists
List I List II
A) \(\Large f(x) = \frac{|x+2|}{x+2}, x \neq -2\) I) \(\Large [\frac{1}{3}, 1]\)
B) \(\large g(x) = |[x]|, x \ \epsilon \) â„ II) Z
C) \(\large h(x) = |x-[x]|, x \ \epsilon\) â„ III) W
D) \(\Large f(x) = \frac{1}{2 - sin \ 3x} , x \epsilon\) â„ IV) [0,1)
V) {-1,1}
Please choose your answer from the right side options
A B C D
V III II I
A B C D
III II IV I
A B C D
V III IV I
A B C D
I II III IV
Assertion (A): (1) + (1 + 2 + 4) + (4 + 6 + 9) + ( 9 + 12 + 16) + ... + (81 + 90 + 100) = 1000
Reason (R): \(\large \overset{n} {\underset{r=1} \Sigma } (r^3 - (r - 1)^3 ) = n^3\) for any natural number n .
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Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true and (R) is not the correct explanation of (A).
(A) is true but (R) is false.
(A) is false but (R) is true.
If A = \(\large \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \) , P = \(\large \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \) and \(\large X= APA^T\) , then \(\large A^T X^{50} A =\)
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\(\large \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
\(\large \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix} \)
\(\large \begin{bmatrix} 25 & 1 \\ 1 & -25 \end{bmatrix} \)
\(\large \begin{bmatrix} 1 & 50 \\ 0 & 1 \end{bmatrix} \)
If [x] is the greatest integer less than or equal to x and |x| is the modulus of x, then the system of three equations
2x +3|y|+ 5[z] = 0, x + |y| - 2[z] = 4, x+ |y| + [z] =1 has,
Please choose your answer from the right side options
a unique solution
finitely many solutions
infinitely many solutions
no solutions
Investigate the λ and μ for the system x + 2y + 3z =6, x + 3y + 5z = 9, 2x + 5y + λz = μ and match the following values in List I with the items in List II
List I List II
A) \(\large \lambda = 8, \mu \neq 15\) I) Infinitely many solutions
B) \(\large \lambda \neq 8, \mu \ \epsilon \) â„ II) No solutions
C) \(\large \lambda = 8, \mu = 15\) III) Unique solutions
The correct match is
Please choose your answer from the right side options
A B C
I II III
A B C
II III I
A B C
III I II
A B C
III II I
If \(\large z = x \ + \ iy, \ x,y \ \epsilon R, \ (x,y) \neq (0, -4)\) and \(\Large Arg(\frac{2z-3}{z+4i}) = \frac{\pi}{4}\) , then the locus of z is
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\(\large 2x^2 + 2y^2 + 5x +5y - 12=0\)
\(\large 2x^2 - 3xy + y^2 + 5x + y - 12=0\)
\(\large 2x^2 + 3xy + y^2 + 5x + y + 12=0\)
\(\large 2x^2 + 2y^2- 11x + 7y - 12 = 0\)
If \(\large z = x+iy, \ x,y \ \epsilon R\) and the imaginary part of \(\Large \frac{\bar z -1 }{\bar z - i}\) is 1 then locus of z is
Please choose your answer from the right side options
x + y + 1 =0
\(\large x + y+1=0, (x,y) \neq (0,-1)\)
\(\large x^2 + y^2 - x + 3y +2 = 0\)
\(\large x^2 + y^2 - x + 3y +2 = 0, (x,y) \neq (0,-1)\)
If ω represents a complex cube root of unity, then \(\Large (1+\frac{1}{\omega})(1+\frac{1}{\omega^2}) + (2+\frac{1}{\omega})(2+\frac{1}{\omega^2})+ ... + (n+\frac{1}{\omega})(n+\frac{1}{\omega^2}) =\)
Please choose your answer from the right side options
\(\Large \frac{n(n^2 +1)}{3}\)
\(\Large \frac{n(n^2 +2)}{3}\)
\(\Large \frac{n(n^2 -2)}{3}\)
\(\Large \frac{n^2(n-1)}{6}\)
If \(tan\,20^{\circ}=\lambda \) , then \(\frac{tan\,160^{\circ}-tan\,110^{\circ}}{1+(tan\,160^{\circ})(tan\,110^{\circ})}= \)
Please choose your answer from the right side options
\(\Large\frac{1+\lambda ^2}{2\lambda } \)
\(\Large \frac{1+\lambda ^2}{\lambda } \)
\(\Large\frac{1-\lambda ^2}{\lambda } \)
\(\Large\frac{1-\lambda ^2}{2\lambda }\)
Consider the circle \(x^2+y^2-6x+4y=12 \). The equation of a tangent to this circle that is parallel to the line \(4x+3y+5=0 \) is
Please choose your answer from the right side options
\(4x +3y+ 10= 0 \)
\(4x+3y-9=0 \)
\(4x+3y+9=0 \)
\(4x+3y-31=0\)
The mean deviation from the mean 10 of the data 6,7,10,12,13, \( \alpha\), 12,16 is
Please choose your answer from the right side options
3.5
3.25
3
3.75
Match the following.
| List-I | List-II |
| \(\int_{-1}^{1}x\left | x \right |dx \) | \(\frac{\pi}{2} \) |
| \(\int_{0}^{\frac{\pi}{2}}\left ( 1+log\left ( \frac{4+3sinx}{4+3cosx} \right ) \right )dx \) | \( \int_{0}^{\frac{a}{2}}f(x)dx \) |
| \(\int_{0}^{a}f(x)dx \) | \( \int_{0}^{a}[f(x)+f(-x)]dx \) |
| \(\int_{-a}^{a}f(x)dx \) | \( 0 \) |
| \( \int_{0}^{a}f(a-x)dx \) |
Please choose your answer from the right side options
I II III IV
d a e c
I II III IV
d a c b
I II III IV
d c a e
I II III IV
a d b d
If f is differentiable,\(f(x+y)=f(x)f(y) \) for all \(x,y\,\epsilon R,f(3)=3,f'(0)=11,\) then\(f'(3)=\)
Please choose your answer from the right side options
\(\frac{3}{11}\)
\(\frac{11}{3}\)
8
33
\(\Large \int_{0}^{\pi }\frac{xdx}{4cos^2x+9sin^2x}= \)
Please choose your answer from the right side options
\(\frac{\pi^2}{12}\)
\(\frac{\pi^2}{4}\)
\(\frac{\pi^2}{6}\)
\(\frac{\pi^2}{3}\)
The probability distribution of a random variable X is given below.
| x =k | 0 | 1 | 2 | 3 | 4 |
| P(X=k) | 0.1 | 0,4 | 0.3 | 0.2 | 0 |
The variance of X is
Please choose your answer from the right side options
1.6
0.24
0.84
0.75
If \(A=\begin{bmatrix} 1 &0 &1 \\ 0& 2& 0\\ 1&-1 &4 \end{bmatrix},A=B+C,B=B^{T}\,\) and \(C=-C^{T}\,\) , then \(C= \)
Please choose your answer from the right side options
\(\begin{bmatrix} 0 &0.5 & 0\\ -0.5 &0 &0 \\ 0 & 0 & 0 \end{bmatrix} \)
\(\begin{bmatrix} 0 &0.5 & 0\\ 0 &0 &0.5 \\ 0 & -0.5 & 0 \end{bmatrix} \)
\(\begin{bmatrix} 0 &-0.5 & 0.5\\ 0.5 &0 &0 \\ -0.5 & 0 & 0 \end{bmatrix} \)
\(\begin{bmatrix} 0 &0.5 & 0\\ -0.5 &0 &0.5 \\ 0 & -0.5 & 0 \end{bmatrix}\)
If \(\vec{a} \) is a unit vector, then \(\left |\vec{a}\times \hat{i} \right |^2+\left |\vec{a}\times \hat{j} \right |^2+\left |\vec{a}\times \hat{k} \right |^2=\)
Please choose your answer from the right side options
2
4
1
0
Let \(f:\mathbb{R}\rightarrow \mathbb{R},\,g:\mathbb{R}\rightarrow \mathbb{R}\) be differentiable function such that \((fog)(x)=x\) . If \(f(x)=2x+cosx+sin^2x\) , then the value of \( \sum_{n=1}^{99}g(1+(2n-1)\pi )\) is
Please choose your answer from the right side options
\(1250\pi\)
\((99)^2\frac{\pi}{2}\)
\((99)^2{\pi}\)
\(2500\pi\)
If \(f:[1,\infty )\rightarrow [1,\infty ]\) is defined by \(f(x)=\frac{1+\sqrt{1+4log_2\,x}}{1}\) then \( \,\,f^{-1}(3)=\)
Please choose your answer from the right side options
\(0\)
\(1\)
\(64\)
\(\frac{1+\sqrt{5}}{2}\)
If \(α\) and \(β\) are the greatest divisors of \(n(n^2-1)\) and \(2n(n^2+2)\) respectively for all \(n\,\epsilon\, N\) then \(αβ=\)
Please choose your answer from the right side options
\(18\)
\(36 \)
\(27 \)
\(9\)
Let \(A=\begin{bmatrix} \frac{1}{6} & \frac{-1}{3} &\frac{-1}{6} \\ \frac{-1}{3}& \frac{2}{3} &\frac{1}{3} \\ \frac{-1}6{} & \frac{1}{3} & \frac{1}{6} \end{bmatrix}\) . If \(A^{2016l}+A^{2017m}+A^{2018n}=\frac{l}{\alpha }A \,\, \) for every \(l,m,n\,\epsilon\, N\) , then the value of \(α\) is
Please choose your answer from the right side options
\(\frac{1}{6}\)
\(\frac{1}{3}\)
\(\frac{1}{2}\)
\(\frac{2}{3}\)
Let \(l,m,n\,\epsilon\, \mathbb{R}\) and \(A=\begin{bmatrix} 1 & r& r^2 &1 \\ r& r^2 &1 & m\\ r^2& 1 &r & n \end{bmatrix}\) . Then the set of all real values of r for which the rank of A is 3, is
Please choose your answer from the right side options
\((0,\infty)\)
\(R\)
\(R-\left \{ 1 \right \}\)
\(R-\left \{ 0 \right \}\)
The following system of equations
\(x+y+z=9 \)
\(2x+5y+7z=52\)
\(x+7y+11z=77\)
has
Please choose your answer from the right side options
no solution
exactly 2 solutions
only one solution
infinitely many solutions
\(Z\) is a complex number such that \(\left | Z \right |\leq 2\) and \( \,\,-\frac{\pi }{3}\leq \,amp\,\,Z\leq \frac{\pi }{3}\) .The area of the region formed by locus of \(Z\) is
Please choose your answer from the right side options
\(\frac{2\pi}{3}\)
\(\frac{\pi}{3}\)
\(\frac{4\pi}{3}\)
\(\frac{8\pi}{3}\)
The points on the argand plane is given by \(Z_1=-3+5i,\,Z_2=-1+6i,\,Z_3=-2+8i,\,Z_4=-4+7i\) form a
Please choose your answer from the right side options
parallelogram
rectangle
rhombus
square
When \(n=8,\,(\sqrt{3}+i)^n+(\sqrt{3}-i)^n=\)
Please choose your answer from the right side options
\(−256 \)
\(−128 \)
\(256i \)
\(128i\)
The domain of the function \(\Large f(x)= sin^{-1} [log_4 \ (\frac{x}{4})] + \sqrt{17x - x^2-16}\) is
Please choose your answer from the right side options
[−1, 1]
[1, 4]
[0, 16]
[1, 16]
If \(\large f:[1,\infty) \longrightarrow [0,\infty)\) is given by \(\Large f(x) = x-\frac{1}{x}\)then \(\large f^{-1}(x)=\)
Please choose your answer from the right side options
\(\large x + \sqrt{x^2 + 4}\)
\(\Large \frac{x}{x^2 -1}\)
\(\Large \frac{1}{2}[x + \sqrt{x^2 + 4}]\)
\(\Large \frac{1}{2}[x - \sqrt{x^2 + 4}]\)
If greatest divisor of \(\large 30.5^{2n} + 4.2^{3n}\) is p, \(\large ∀ ∈ N\) and \(\large 2^{2n+1} - 6n -2\) is q, \(\large ∀ ∈ N\), then p + q =
Please choose your answer from the right side options
26
52
104
13
If \(\large A=\begin{bmatrix} 0 & 5 \\ 0 & 0 \\ \end{bmatrix}\) and \(\large f(x) = x+x^2 +...+ \ x^{2018}\), the f(A) + I =
Please choose your answer from the right side options
\(\large \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix}\)
\(\large \begin{bmatrix} 1 & 5 \\ 0 & 0 \\ \end{bmatrix}\)
\(\large \begin{bmatrix} 0 & 5 \\ 1 & 1 \\ \end{bmatrix}\)
\(\large \begin{bmatrix} 1 & 5 \\ 0 & 1 \\ \end{bmatrix}\)
If a, b, c are real numbers such that a−b= 1, b-c = 3 then the number of matrices of the form\(\large A= \begin{bmatrix} 1 & 1 & 1\\ a & b &c \\ a^2 & b^2 & c^2\\ \end{bmatrix}\) such that |A| = -12 is
Please choose your answer from the right side options
1
2
3
Infinitely
For what values of ‘a’ the system of equations x + y + z = 1, 2x + 3y + 2z =2, ax + ay + 2az = 4 will have a unique solution?
Please choose your answer from the right side options
For a = 0 only.
For \(\large a \ \epsilon R - \{0\}\)
For all \(\large a \ \epsilon \ \mathbb{Q}\)
For all \(\large a \notin \mathbb{N}\)
If \(\large z_n =(1+i\sqrt 2)^{n}, n \ \epsilon \ \mathbb{Z}\), then \(\Large \frac{1}{9} Re(z_4 \bar z_5) = \)
Please choose your answer from the right side options
81
27
9
3
If z = x + iy, then the centre of circle \(\Large |\frac{z- 3}{z-2i} | = 2\) is
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\(\Large (-1, -\frac{8}{3})\)
\(\Large (1, \frac{8}{3})\)
\(\Large (-1, \frac{8}{3})\)
\(\Large (1, -\frac{8}{3})\)
If z is a complex number such that \(\large |z + 4| \geq 3,\) then the smallest value of \(\large |z+3|\) is
Please choose your answer from the right side options
3
1
2
0
Match the items of List-I with those of the items of List-II
| List-I | List-II |
| A) Range of \(Sec^{-1}[1+cos^2x],[.] \) denote greatest integer function | I) odd function |
| B) Domain of f(x), where \(f\left ( x+{\Large\frac{1}{x}} \right ) =x^2+{\Large\frac{1}{x^2}} \) | II)\(\left\{ 0,\Large\frac{1}{2}\right\}\) |
| C) \(f(x+y)=f(x)+f(y) :f(1)=5\) | III)\(\left\{ Sec^{-1}5,Sec^{-1}4\right\} \) |
| D)\(Sin^{-1}x-Cos^{-1}x+Sin^{-1}(1-x)=0\Rightarrow x\,\epsilon\) | IV) \(\mathbb{R}-(-2,2)\) |
| V) \(\left\{ {Sec^{-1}1 ,Sec^{-1}2}\right\}\) |
Please choose your answer from the right side options
A B C D
V IV I II
A B C D
III IV II I
A B C D
V II III IV
A B C D
III II I IV
The domain of the function \(f(x)=Sec^{-1}(3x-4)+Tanh^{-1}\left ( {\Large\frac{x+3}{5}} \right ) \) is
Please choose your answer from the right side options
\((-8,1)\cup \left ( {\Large\frac{5}{3}},2 \right ) \)
\(\left (1, {\Large\frac{5}{3}} \right ) \)
\([-8,1] \cup \left [ {\Large\frac{5}{3}},2 \right ]\)
\((-8,1]\cup \left [ {\Large\frac{5}{3}},2 \right )\)
Let \(m = (9n^2 + 54n + 80) ( 9n^2 + 45n + 54) ( 9n^2 + 36n + 35)\). The greatest positive integer which divides m, for all positive integers n , is
Please choose your answer from the right side options
720
724
696
842
If A is a 3×3 matrix and the matrix obtained by replacing the elements of A with their corresponding cofactors is \(\begin{bmatrix} 1& -2 & 1\\ 4& -5& -2 \\ -2& 4&1 \\ \end{bmatrix}\) , then a possible value of the determinant of A is
Please choose your answer from the right side options
4
3
2
1
If \(A=\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \\ \end{bmatrix} \) then \(\left ( A^2 \right )^{-1} \) =
Please choose your answer from the right side options
\(A^2\)
2A
\(A^3\)
\( A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \\ \end{bmatrix}\)
The equations x + y + z = 3, x + 2y + 2z = 6 and x + ay +3z = b have
Please choose your answer from the right side options
No solution when a ≠ 3 , b is any value
Infinite number of solutions when b ≠ 9
Unique solution when a ≠ 3, b is any value
Unique solution when a = 3 and b ≠ 9
Let \(Z\,\epsilon\,\mathbb{C}\) and \(i=\sqrt{-1} \), if \(a,b,c\,\epsilon (0,1) \) be such that \(a^2+b^2+c^2=1 \) and \(b+ic=(1+a)z\) , then \(\Large\frac{1+iz}{1-iz}\)=
Please choose your answer from the right side options
\(\Large\frac{a+ib}{1+c} \)
\(\Large\frac{a-ib}{1+c} \)
\(\Large\frac{a-ib}{1-c} \)
\(\Large\frac{a+ib}{1-c}\)
If \(A=\begin{Bmatrix} z=x+iy/\,real\,\,part\,\,of\,\,{\Large\frac{\bar{z}-1}{z-i}}=2\end{Bmatrix}\), then the locus of the point \(P( x, y)\) in the cartesian plane is
Please choose your answer from the right side options
a pair of lines passing through (-1, -1)
a circle of radius \(\Large\frac{1}{\sqrt2}\) and the centre \(\left (\Large \frac{-1}{2},\frac{3}{2} \right )\)
a pair of lines passing through (-1, -2)
a circle of radius \(\Large\frac{1}{2}\)
If \(ω\) is a complex cube root of unity ,then \(\left ( 1-\omega +\omega^2 \right )^6+\left ( 1-\omega^2+\omega \right )^6\) =
Please choose your answer from the right side options
0
6
64
128
Matrix \(A_r= \begin{bmatrix} r & r-1\\ r-1 & r \end{bmatrix} ; r=1,2,3,.....\) If \( \sum_{r=1}^{100}\begin{vmatrix} A_r \end{vmatrix}=(\sqrt{10})^k\) , then \(K=..........;(\left | A_r \right |=det(A_r))\)
Please choose your answer from the right side options
\(2\)
\(6\)
\(4\)
\(8\)
\(\frac{d}{dx}\left ( 3cos\left ( \frac{\pi }{6}+x^\circ \right ) -4cos^3\left ( \frac{\pi}{6} +x^\circ\right )\right )=................\)
Please choose your answer from the right side options
\(cos(3x^\circ)\)
\(\frac{\pi}{60}sin(3x^\circ)\)
\(\frac{\pi}{60}cos(3x^\circ)\)
\(-\frac{\pi}{60}sin(3x^\circ)\)
If \(f(x)=1+x+x^2+.......x^{1000}\) , then \( f'(–1) = …….\)
Please choose your answer from the right side options
\( –50 \)
\(–500 \)
\( –100 \)
\(500500 \)
Applying mean value theorem on \(f(x)=logx;\,x\,\varepsilon\, [1,e]\) the value of c = ………
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\(log(e-1)\)
\(e-1\)
\(1-e\)
\(2\)
If \(\int sin^{13}\,xcos^3\,xdx=Asin^{14}x+Bsin^{16}x+C\) , then A + B = ……..
Please choose your answer from the right side options
\(\frac{1}{110}\)
\(\frac{17}{112}\)
\(\frac{15}{112}\)
\(\frac{1}{112}\)
If \(\int \frac{1+cosx}{cosx-cos^2x}dx=log\left | secx+tanx \right |-2f'(x)+C\) , then f(x) = ……..
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\(2\,cot\left ( \frac{x}{2} \right )\)
\(2\,log\left | sin\frac{x}{2} \right |\)
\(-2\,cot\left ( \frac{x}{2}\right )\)
\(-2\,log\left | sin\frac{x}{2} \right |\)
The probability that an event A occurs in a single trial of an experiment is \(0.6\) . In the first three independent trials of the experiment, the probability that A occurs atleast once is ……….
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\( 0.930 \)
\( 0.936\)
\( 0.925 \)
\( 0.927\)
If \(6P(A)=8P(B)=14P(A\cap B)=1\), then the \(P\left ( \frac{A'}{B} \right )=.........\)
Please choose your answer from the right side options
\(\frac{3}{7}\)
\(\frac{4}{7}\)
\(\frac{3}{5}\)
\(\frac{2}{5}\)
The mean and variance of a random variable X having a binomial distribution are 6 and 3 respectively. The probability of variable X less than 2 is
Please choose your answer from the right side options
\(\frac{13}{2048}\)
\(\frac{13}{4096}\)
\(\frac{15}{4096}\)
\(\frac{25}{2048}\)
\(S_n\) denotes the sum of n terms of an AP, whose first term is a. If the common difference \(d = S_n– k \,\,S_{n-1} + S_{n–2}\) , then k is equal to
Please choose your answer from the right side options
2
3
5
7
If \(Z_1 \) and \(Z_2 \) are two complex numbers such that \(\left|Z_1 \right| =\left|Z_2 \right| \) and \(arg(Z_1) +arg(Z_2) =\pi \), then \(Z_1 \) is equal to
Please choose your answer from the right side options
\(2\bar{Z_2} \)
\(\bar{Z_2} \)
\(-\bar{Z_2}\)
None of these
If \(Z_1, Z_2\) and \(Z_3\) represent the vertices of an equilateral triangle such that \(|Z_1| = |Z_2| = |Z_3|\), then
Please choose your answer from the right side options
\(Z_1 + Z_2 = Z_3\)
\(Z_1 + Z_2 + Z_3 = 0\)
\(Z_1 Z_2 = Z_3\)
\(Z_1 – Z_2 = Z_3 – Z_2\)
If the equation \(x^2+2x+3=0\) and \(ax^2+bx+c=0\), a, b, c ∈ R, have a common root, then a:b:c is
Please choose your answer from the right side options
3 : 2 : 1
1:3:2
3:1:2
1:2:3
If a, b, c are in GP and \(a^{\Large\frac{1}{x}}=b^{\Large\frac{1}{y}}=c^{\Large\frac{1}{z}}\) , then x, y, z are in
Please choose your answer from the right side options
AP
GP
HP
None of these
If p, q, r and s are positive real numbers such that p + q + r + s = 2, then M = (p+q) (r+s) satisfies the relation, when
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0 < M \(\leq\) 1
1 \(\leq\) M \(\leq\) 2
2 \(\leq\) M \(\leq\) 3
3 \(\leq\) M \(\leq\) 4
The sum of the infinite series \({\Large\frac{2^2}{2!}}+{\Large\frac{2^4}{4!}}+{\Large\frac{2^6}{6!}}+...... \) is
Please choose your answer from the right side options
\({\Large\frac{e^2+1}{2}} \)
\({\Large\frac{e^4+1}{2e^2}} \)
\({\Large\frac{(e^2-1)^2}{2e^2}} \)
\({\Large\frac{(e^2+1)^2}{2e^2}} \)
If n is a positive integer, then \(n^3 + 2n\) is divisible by
Please choose your answer from the right side options
2
6
15
3
If a and b are the coefficients of \(x^r\) and \(x^{n–r}\) respectively in the expansion of \((1+x)^n\), then
Please choose your answer from the right side options
a = b
\(a + b = n^2\)
a = nb
a – b = n
If \(f\) is differentiable at \(x=1\) , then \(\lim_{x\rightarrow 1}\frac{x^2f(1)-f(x)}{x-1}\) is
Please choose your answer from the right side options
\(-f'(1)\)
'>\(f(1)-f'(1)\)
'>\(2f(1)-f'(1)\)
'>\(2f(1)+f'(1)\)
'>\(f(1)+f'(1)\)
Eccentricity of the ellipse \(4x^2+y^2-8x+4y-8=0\) is
Please choose your answer from the right side options
\(\frac{\sqrt3}{2}\)
\(\frac{\sqrt3}{4}\)
\(\frac{\sqrt3}{\sqrt2}\)
\(\frac{\sqrt3}{8}\)
\(\frac{\sqrt3}{16}\)
The focus of the parabola \((y+1)^2=-8(x+2)\) is
Please choose your answer from the right side options
\((-4,-1)\)
\((-1,-4)\)
\((1,4)\)
\((4,1)\)
\((-1,4)\)
Which of the following is the equation of a hyperbola?
Please choose your answer from the right side options
\(x^2-4x+16y+17=0\)
\(4x^2+4y^2-16x+4y-60=0\)
\(x^2+2y^2+4x+2y-27=0\)
\(x^2-y^2+3x-2y-43=0\)
\(x^2+4x+6y-2=0\)
Let \(f(x)=px^2+qx+r\) , where \(p,q,r\) are constants and \(p\neq0\) If \(f(5)=-3f(2)\) and \(f(-4)=0\) , then the other root of \(f\) is
Please choose your answer from the right side options
\(3\)
\(-7\)
\(-2\)
\(2\)
\(6\)
Let \(f:R\rightarrow R\) satisfy \(f(x)f(y)=f(xy)\) for all real numbers \(x\) and \(y\) . If \(f(2)=4\) , then \(f(\frac{1}{2})=\)
Please choose your answer from the right side options
\(0\)
\(\frac{1}{4}\)
\(\frac{1}{2}\)
\(1\)
\(2\)
Sum of last \(30\) coefficents in the binomial expansion of \((1+x)^{59}\) is
Please choose your answer from the right side options
\(2^{29}\)
\(2^{59}\)
\(2^{58}\)
\(2^{59}-2^{29}\)
\(2^{60}\)
\((\sqrt{3}+\sqrt{2})^4-(\sqrt{3}-\sqrt{2})^4=\)
Please choose your answer from the right side options
\(20\sqrt6\)
\(30\sqrt6\)
\(5\sqrt{10}\)
\(40\sqrt6\)
\(10\sqrt6\)
Three players \(A,B\) and \(C\) play a game. The probability that \(A,B\) and \(C\) will finish the game are respectively \(\frac{1}{2},\frac{1}{3}\) and \(\frac{1}{4}\) . The probability that the game is finished is.
Please choose your answer from the right side options
\(\frac{1}{8}\)
\(1\)
\(\frac{1}{4}\)
\(\frac{3}{4}\)
\(\frac{1}{2}\)
If \(z_1=2-i\) and \(z_2=1+i\) , then \(\large \left | \frac{z_1+z_2+1}{z_1-z_2+i} \right |\) is
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\(2\)
\(2\sqrt2\)
\(3\)
\(\sqrt3\)
None of these
If \(f(x)=\large \sqrt{\frac{x-sinx}{x+cos^2x}}\), then \(\lim_{x\rightarrow \infty }f(x)\) is equal to
Please choose your answer from the right side options
\(1\)
\(2\)
\(\frac{1}{2}\)
\(0\)
\(\infty\)
The value of \(sin\frac{31}{3}\pi\) is
Please choose your answer from the right side options
\(\frac{\sqrt3}{2}\)
\(\frac{1}{\sqrt2}\)
\(\frac{-\sqrt3}{2}\)
\(\frac{-1}{\sqrt2}\)
\(\frac{1}{2}\)
The sum of odd integers from \(1\) to \(2001\) is
Please choose your answer from the right side options
\((1121)^2\)
\((1101)^2\)
\((1001)^2\)
\((1021)^2\)
\((1011)^2\)
If \(y=\frac{sin^2x}{1+cotx}+\frac{cos^2x}{1+tanx}\) , then \(y'(x)\) is equal to
Please choose your answer from the right side options
\(2cos^2x\)
\(2cos^3x\)
\(-cos2x\)
\(cos2x\)
\(3cosx\)
The foci of the hyperbola \(16x^2-9y^2-64x+18y-90=0\) are
Please choose your answer from the right side options
\(\left ( \frac{24\pm 5\sqrt{145}}{12} ,1\right )\)
\(\left ( \frac{21\pm 5\sqrt{145}}{12} ,1\right )\)
\(\left ( 1,\frac{24\pm 5\sqrt{145}}{2} \right )\)
\(\left ( 1,\frac{21\pm 5\sqrt{145}}{2} \right )\)
\(\left ( \frac{21\pm 5\sqrt{145}}{2} ,-1\right ) \)
If the sum of the coefficients in the expansions of \((a^2x^2-2ax+1)^{51}\) is zero, then \(a\) is equal to
Please choose your answer from the right side options
\(0\)
\(1\)
\(-1\)
\(-2\)
\(2\)
The mean deviation of the data \(2,9,9,3,6,9,4\) from the mean is
Please choose your answer from the right side options
\(2.23\)
\(3.23\)
\(2.57\)
\(3.57\)
\(1.03\)
The mean and variance of a binomial distribution are \(8\) and \(4\) respectively. What is \((X=1)\) ?
Please choose your answer from the right side options
\(\frac{1}{2^8}\)
\(\frac{1}{2^{12}}\)
\(\frac{1}{2^6}\)
\(\frac{1}{2^4}\)
\(\frac{1}{2^5}\)
The number of diagonals of a polygon with \(15\) sides is
Please choose your answer from the right side options
\(90\)
\(45\)
\(60\)
\(70\)
\(10\)
In a class, \(40\%\) of students study Maths and Science and \(60\%\) of students study Maths. What is the probability of a students studying Science given the student is already studying Maths?
Please choose your answer from the right side options
\(\frac{1}{3}\)
\(\frac{1}{6}\)
\(\frac{2}{3}\)
\(\frac{1}{5}\)
\(\frac{1}{4}\)
The eccentricity of the conic \(x^2+2y^2-2x+3y+2=0\) is
Please choose your answer from the right side options
\(0\)
\(\frac{1}{\sqrt2}\)
\(\frac{1}{2}\)
\(\sqrt2\)
\(1\)
If the mean of a set of observations \(x_1,x_2,...x_{10}\) is \(20\) , then the mean of \(x_1+4,x_2+8,x_3+12,...x_{10}+40\) is
Please choose your answer from the right side options
\(34\)
\(32\)
\(42\)
\(38\)
\(40\)
A letter is taken at random from the word “STATISTICS” and another letter is taken at random from the word “ASSISTANT”. The probability that they are same letters is
Please choose your answer from the right side options
\(\frac{1}{45}\)
\(\frac{13}{90}\)
\(\frac{19}{90}\)
\(\frac{5}{18}\)
\(\frac{9}{10}\)
If \(sin\,\alpha\) and \(cos\,\alpha\) are the roots of the equation \(ax^2+bx+c=0\) , then
Please choose your answer from the right side options
\(a^2-b^2+2ac=0\)
\((a-c)^2=b^2+c^2\)
\(a^2+b^2-2ac=0\)
\(a^2+b^2+2ac=0\)
\(a+b+c=0\)
If the sides of triangle are \(4,5\) and \(6\) cm. Then the area (in sq cm) of triangle is
Please choose your answer from the right side options
\(\frac{\pi}{4}\)
\(\frac{\pi}{4} \sqrt7\)
\(\frac{4}{15}\)
\(\frac{4}{15}\sqrt7\)
\(\frac{15}{4}\sqrt7\)
In a group of \(6\) boys and \(4\) girls, a team consisting of four children is formed such that the team has atleast one boy. The number of ways of forming a team like this is
Please choose your answer from the right side options
\(159\)
\(209\)
\(200\)
\(240\)
\(212\)
A password is set with \(3\) distinct letters from the word LOGARITHMS. How many such passwords can be formed?
Please choose your answer from the right side options
\(90\)
\(720\)
\(80\)
\(72\)
\(120\)
If \(5^{97}\) is divided by \(52\) , the remainder obtained is
Please choose your answer from the right side options
\(3\)
\(5\)
\(4\)
\(0\)
\(1\)
A quadratic equation \(ax^2+bx+c=0\) , with distinct coefficients is formed. It \(a,b,c\) are chosen from the numbers \(2,3,5\) then the probability that the equation has real roots is
Please choose your answer from the right side options
\(\frac{1}{3}\)
\(\frac{2}{5}\)
\(\frac{1}{4}\)
\(\frac{1}{5}\)
\(\frac{2}{3}\)
\(\lim_{x\rightarrow \infty }\frac{3x^3+2x^2-7x+9}{4x^3+9x-2}\) is equal to
Please choose your answer from the right side options
\(\frac{2}{9}\)
\(\frac{1}{2}\)
\(\frac{-9}{2}\)
\(\frac{3}{4}\)
\(\frac{9}{2}\)
The minimum value of \(f(x)=max\left \{ x,1+x,2-x \right \}\) is
Please choose your answer from the right side options
\(\frac{1}{2}\)
\(\frac{3}{2}\)
\(1\)
\(0\)
\(2\)
The equations of the asymptotes of the hyperbola \(xy+3y-2y-10=0\) are
Please choose your answer from the right side options
\(x=-2,y=-3\)
\(x=2,y=-3\)
\(x=2,y=3\)
\(x=4,y=3\)
\(x=3,y=4\)
If \(f(x)=x^6+6^x\), then \(f'(x)\) is equal to
Please choose your answer from the right side options
\(12x\)
\(x+4\)
\(6x^5+6^xlog(6)\)
\(6x^5+x6^{x-1}\)
\(x^6\)
The standard deviation of the data \(6,5,9,13,12,8,10\) is
Please choose your answer from the right side options
\(\frac{\sqrt{52}}{7}\)
\(\frac{52}{7}\)
\(\frac{\sqrt{53}}{7}\)
\(\frac{53}{7}\)
\(6\)
\(\lim_{x\rightarrow 0}\frac{1-cos\,mx}{1-cos\,nx}=\)
Please choose your answer from the right side options
\(\frac{m^2}{n^2}\)
\(\frac{n^2}{m^2}\)
\(\infty\)
\(-\infty\)
\(0\)
\(\lim_{x\rightarrow 0}\frac{(\sqrt{1+2x})-1}{x}=\)
Please choose your answer from the right side options
\(0\)
\(-1\)
\(\frac{1}{2}\)
\(1\)
\(\frac{-1}{2}\)
Let \(f\) and \(g\) be differentiable functions such that \(f(3)=5,g(3)=7,f'(3)=13,g'(3)=6,f'(7)=2\,\,and\,\,g'(7)=0\) . If \(h(x)=(fog)(x)\) , then \(h'(3)=\)
Please choose your answer from the right side options
\(14\)
\(12\)
\(16\)
\(0\)
\(10\)
\(\frac{\sqrt{3}}{sin(20^\circ)}-\frac{1}{cos(20^\circ)}=\)
Please choose your answer from the right side options
\(1\)
\(\frac{1}{\sqrt2}\)
\(2\)
\(4\)
\(0\)
A poison variate \(X\) satisfies \(P(X=1)=P(X=2).P(X=6)\) is equal to
Please choose your answer from the right side options
\(\frac{4}{45}\,e^{-2}\)
\(\frac{1}{45}\,e^{-1}\)
\(\frac{1}{9}\,e^{-2}\)
\(\frac{1}{4}\,e^{-2}\)
\(\frac{1}{45}\,e^{-2}\)
Let \(a\) and \(b\) be \(2\) consecutive integers selected from the first \(20\) natural numbers. The probability that \(\sqrt{a^2+b^2+a^2b^2}\) is an odd positive integer is
Please choose your answer from the right side options
\(\frac{9}{19}\)
\(\frac{10}{19}\)
\(\frac{13}{19}\)
\(1\)
\(0\)
An ellipse of eccentricity \(\frac{2\sqrt2}{3}\) is inscribed in a circle. A point is chosen inside the circle at random. The probability that the point lies outside the ellipse is
Please choose your answer from the right side options
\(\frac{1}{3}\)
\(\frac{2}{3}\)
\(\frac{1}{9}\)
\(\frac{2}{9}\)
\(\frac{1}{27}\)
If the vectors \(4\hat{i}+11\hat{j}+m\hat{k},7\hat{i}+2\hat{j}+6\hat{k}\,\,and\,\,\hat{i}+5\hat{j}+4\hat{k}\) are coplanar, then \(m\) is equal to
Please choose your answer from the right side options
\(38\)
\(0\)
\(10\)
\(-10\)
\(25\)
Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=\hat{i}+3\hat{j}+5\hat{k}\,\,and\,\,\vec{c}=7\hat{i}+9\hat{j}+11\hat{k}\) . Then, the area of the parallelogram with diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is
Please choose your answer from the right side options
\(4\sqrt6\)
\(\frac{1}{2}\sqrt{21}\)
\(\frac{\sqrt6}{2}\)
\(\sqrt6\)
\(\frac{1}{\sqrt6}\)
If \(\left | \vec{a} \right |=3,\left | \vec{b} \right |=1,\left | \vec{c} \right |=4\) and \(\vec{a}+\vec{b}+\vec{c}=0\) , then the value of \(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}\) is equal to
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\(13\)
\(26\)
\(-29\)
\(-13\)
\(-26\)
If \(\left | \vec{a}-\vec{b} \right |=\left | \vec{a} \right |=\left | \vec{b} \right |=1\) , then the angle between \(\vec{a}\) and \(\vec{b}\) is equal to
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\(\frac{\pi}{3}\)
\(3\frac{\pi}{4}\)
\(\frac{\pi}{2}\)
\(0\)
\(\pi\)
If the vectors \(\vec{a}=\hat{i}-\hat{j}+2\hat{k},\vec{b}=2\hat{i}+4\hat{j}+\hat{k}\,\,and\,\,\vec{c}=\lambda \hat{i}+9\hat{j}+\mu \hat{k}\) are mutually orthogonal, then \(\lambda+\mu\) is equal to
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\(5\)
\(-9\)
\(-1\)
\(0\)
\(-5\)
The solution of \(x^{\frac{2}{5}}+3x^{\frac{1}{5}}-4=0\) are
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\(1,1024\)
\(-1,1024\)
\(1,1031\)
\(-1024,1\)
\(-1,1031\)
If the equations \(x^2+ax+1=0\) and \(x^2-x-a=0\) have a real common root \(b\) , then the value of \(b\) is equal to
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\(0\)
\(1\)
\(-1\)
\(2\)
\(3\)
If \(sin\,\theta-cos\,\theta=1\) , then the value of \(sin^3\,\theta-cos^3\,\theta\) is equal to
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\(1\)
\(-1\)
\(0\)
\(2\)
\(-2\)
Two dice of different colours are thrown at a time. The probability that the sum is either \(7\) or \(11\) is
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\(\frac{7}{36}\)
\(\frac{2}{9}\)
\(\frac{2}{3}\)
\(\frac{5}{9}\)
\(\frac{6}{7}\)
\(\frac{1}{9!}+\frac{1}{3!7!}+\frac{1}{5!5!}+\frac{1}{7!3!}+\frac{1}{9!}\) Is equal to
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\(\frac{2^9}{10!}\)
\(\frac{2^{10}}{8!}\)
\(\frac{2^{11}}{9!}\)
\(\frac{2^{10}}{7!}\)
\(\frac{2^8}{9!}\)
The order and degree of the differential equation \((y''')^2+(y'')^3-(y')^4+y^5=0\) is
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\(3\,and\,2\)
\(1\,and\,2\)
\(2\,and\,3\)
\(1\,and\,4\)
\(3\,and\,5\)
\(\int_{-2}^{2}\left | x \right |dx\) is equal to
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\(0\)
\(1\)
\(2\)
\(4\)
\(\frac{1}{2}\)
\(\int_{-1}^{0}\frac{dx}{x^2+2x+2}\) is equal to
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\(0\)
\(\frac{\pi}{4}\)
\(\frac{-\pi}{4}\)
\(\frac{\pi}{2}\)
\(\frac{-\pi}{2}\)
If \(\int_{-1}^{4}f(x)dx=4\) and \(\int_{2}^{4}(3-f(x))dx=7\) , then \(\int_{-1}^{2}f(x)dx\) is
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\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(\int \frac{xe^x}{(1+x)^2}dx=\)
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\( \frac{e^x}{1+x}+C\)
\( \frac{e^x}{1+e^x}+C\)
\( \frac{e^{2x}}{1+e^x}+C\)
\( \frac{e^{-x}}{1+x}+C\)
\( \frac{e^{-2x}}{1+x}+C\)
The remainder when \(2^{2000}\) is divided by \(7\) is
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\(1\)
\(2\)
\(8\)
\(12\)
\(4\)
The coefficient of \(x^5\) in the expansion of \((x+3)^8\) is
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\(1542\)
\(1512\)
\(2512\)
\(12\)
\(4\)
The maximum value of \(5\,cos\,\theta+3\,cos\left ( \theta+\frac{\pi}{3} \right )+3\) is
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\(5\)
\(11\)
\(10\)
\(-1\)
\(2\)
The area of the triangle in the complex plane formed by \(z,iz\) and \(z+iz\) is
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\(\left | z \right |\)
\(\left | \bar{z} \right |^2\)
\(\frac{1}{2}\left | {z} \right |^2\)
\(\frac{1}{2}\left | {z+iz} \right |^2\)
\(\left | {z+iz} \right |\)
Let \(f:f(-x)\rightarrow f(x)\) be a differentiable function. If \(f\) is even, then \(f'(0)\) is equal to
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\(1\)
\(2\)
\(0\)
\(-1\)
\(\frac{1}{2}\)
The coordinate of the point dividing internally the line joining the points \((4,-2)\) and \((8,6)\) in the ratio \(7:5\) is
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\((16,18)\)
\((18,16)\)
\((\frac{19}{3},\frac{8}{3})\)
\((\frac{8}{3},\frac{19}{3})\)
\((7,3)\)
The area of the triangle formed by the points \((a,b+c),(b,c+a),(c,a+b)\) is
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\(abc\)
\(a^2+b^2+c^2\)
\(ab+bc+ca\)
\(0\)
\(a(ab+bc+ca)\)
If \((x,y)\) is equidistant from \((a+b,b-a)\) and \((a-b,b+a)\), then
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\(ax+by=0\)
\(ax-by=0\)
\(bx+ay=0\)
\(bx-ay=0\)
\(x=y\)
The equation of the line passing through \((a,b)\) and parallel to the line \(\frac{x}{a}+\frac{y}{b}=1\) is
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\(\frac{x}{a}+\frac{y}{b}=3\)
\(\frac{x}{a}+\frac{y}{b}=2\)
\(\frac{x}{a}+\frac{y}{b}=0\)
\(\frac{x}{a}+\frac{y}{b}+2=0\)
\(\frac{x}{a}+\frac{y}{b}=4\)
If the points \((2a,a),(a,2a)\) and \((a,a)\) enclose a triangle of area \(18\,sq\) units, then the centroid of the triangle is equal to
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\((4,4)\)
\((8,8)\)
\((-4,-4)\)
\((4\sqrt{2},4\sqrt{2})\)
\((6,6)\)
The area of a triangle is \(5\,sq\) units. Two of its vertices are \((2,1)\) and \((3,-2)\) . The third vertex lies on \(y=x+3\) . The coordinates of the third vertex can be
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\(\left (\frac{-3}{2}, \frac{-3}{2} \right )\)
\(\left (\frac{3}{4}, \frac{-3}{2} \right )\)
\(\left (\frac{7}{2}, \frac{13}{2} \right )\)
\(\left (\frac{-1}{4}, \frac{1}{2} \right )\)
\(\left (\frac{3}{2}, \frac{3}{2} \right )\)
If \(x^2+y^2+2gx+2fy+1=0\) represents a pair of straight lines, then \(f^2+g^2\) is equal to
Please choose your answer from the right side options
\(0\)
\(1\)
\(2\)
\(4\)
\(3\)
If \(\theta\) is the angle between the pair of straight lines \(x^2-5xy+4y^2+3x-4=0\) , then \(tan^2\,\theta\) is equal to
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\(\frac{9}{16}\)
\(\frac{16}{25}\)
\(\frac{9}{25}\)
\(\frac{21}{25}\)
\(\frac{25}{9}\)
If \(3\hat{i}+2\hat{j}-5\hat{k}=x(2\hat{i}-\hat{j}+\hat{k})+y(\hat{i}+3\hat{j}-2\hat{k})+z(-2\hat{i}+\hat{j}-3\hat{k})\), then
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\(x=1,y=2,z=3\)
\(x=2,y=3,z=1\)
\(x=3,y=1,z=2\)
\(x=1,y=3,z=2\)
\(x=2,y=2,z=3\)
\(sin\,15^{\circ}=\)
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\(\frac{\sqrt{3}-1}{2\sqrt{2}}\)
\(\frac{\sqrt{3}+1}{2\sqrt{2}}\)
\(\frac{1-\sqrt{3}}{2\sqrt{2}}\)
\(\frac{1+\sqrt{3}}{2\sqrt{2}}\)
\(\frac{-(1+\sqrt{3})}{2\sqrt{2}}\)
If \(\bar{a}\) and \(\bar{b}=3\hat{i}+6\hat{j}+6\hat{k}\) are collinear and \(\bar{a}\cdot \bar{b}=27\), then \(\bar{a}\) is equal to
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\(3(\hat{i}+\hat{j}+\hat{k})\)
\(\hat{i}+2\hat{j}+2\hat{k}\)
\(2\hat{i}+2\hat{j}+2\hat{k}\)
\(\hat{i}+3\hat{j}+3\hat{k}\)
\(\hat{i}-3\hat{j}+2\hat{k}\)
If \(\left | \vec{a} \right |=13,\left | \vec{b} \right |=5 \,\,and\,\,\vec{a}\cdot \vec{b}=30\) , then \(\left | \vec{a}\times \vec{b} \right |\)is equal to
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\(30\)
\(\frac{30}{25}\sqrt{233}\)
\(\frac{30}{33}\sqrt{193}\)
\(\frac{65}{23}\sqrt{493}\)
\(\frac{65}{13}\sqrt{133}\)
If \(^{56}P_{r+6}:\,^{54}P_{r+3}=30800:1\) , then \(r\) is equal to
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\(69\)
\(41\)
\(51\)
\(61\)
\(49\)
Distance between two parallel lines \(y=2x+4\) and \(y=2x-1\) is
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\(5\)
\(5\sqrt5\)
\(\sqrt5\)
\(\frac{1}{5}\)
\(\frac{3}{\sqrt5}\)
\((^7C_0\,+\,^7C_1)+(^7C_2\,+\,^7C_3)+...+(^7C_6\,+\,^7C_7)=\)
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\(2^8-2\)
\(2^7-1\)
\(2^7\)
\(2^8-1\)
\(2^7-2\)
The coefficient of \(x\) in the expansion of \((1-3x+7x^2)(1-x)^{16}\) is
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\(17\)
\(19\)
\(-17\)
\(-19\)
\(20\)
The equation of the circle with centre \((2,2)\) which passes through \((4,5)\) is
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\(x^2+y^2-4x+4y-77=0\)
\(x^2+y^2-4x-4y-5=0\)
\(x^2+y^2+2x+2y-59=0\)
\(x^2+y^2-2x-2y-23=0\)
\(x^2+y^2+4x-2y-26=0\)
The point in the \(xy-\) plane which is equidistant from \((2,0,3),(0,3,2)\)and \((0,0,1)\) is
Please choose your answer from the right side options
\((1,2,3)\)
\((-3,2,0)\)
\((3,-2,0)\)
\((3,2,0)\)
\((3,2,1)\)
Let \(f:x\rightarrow y\) be such that \(f(1)=2\) and \(f(x+y)=f(x)f(y)\) for all natural numbers \(x\) and \(y\) . If \(\sum_{k=1}^{n}f(a+k)=16(2^n-1)\) , then \(a\) is equal to
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\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
If \(^nC_{r-1}=36,\,^nC_{r}=84\) and \(^nC_{r+1}=126\) , then \(n=\)
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\(3\)
\(4\)
\(8\)
\(9\)
\(10\)
Let \(f:(-1,1)\rightarrow (-1,1)\) be continuous, \(f(x)=f(x)^2\) for all \(x\,\epsilon \,(-1,1)\) and \(f(0)=\frac{1}{2}\) , then the value of \(4f(\frac{1}{4})\) is
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\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(\lim_{x\rightarrow \infty }\sqrt{x^2+1}-\sqrt{x^2-1}=\)
Please choose your answer from the right side options
\(-1\)
\(1\)
\(0\)
\(2\)
\(4\)
If \(f\) is differentiable at \(x=1\) and \(\lim_{h\rightarrow 0 }\frac{1}{h}f(1+h)=5,f'(1)=\)
Please choose your answer from the right side options
\(0\)
\(1\)
\(3\)
\(4\)
\(5\)
The maximum value of the function \(2x^3-15x^2+36x+4\) is attained at
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\(0\)
\(3\)
\(4\)
\(2\)
\(5\)
If \(\int f(x)\,cos\,xdx=\frac{1}{2}\left \{ f(x) \right \}^2+C\) , then \(f(\frac{\pi}{2})\) is
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\(C\)
\(\frac{\pi }{2}+C\)
\(C+1\)
\(2\pi+C\)
\(C+2\)
\(\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{x}{1+sin\,x}dx=\)
Please choose your answer from the right side options
\(\pi(\sqrt2+2)\)
\(\pi(\sqrt2+1)\)
\(2\pi(\sqrt2-1)\)
\( 2\pi(\sqrt2+1)\)
\(\large \frac{\pi}{\sqrt2+1}\)
\(\int_{0}^{\frac{\pi }{2}}\frac{2^{sin\,x}}{2^{sin\,x}+2^{cos\,x}}dx=\)
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\(2\)
\(\pi\)
\(\frac{\pi}{4}\)
\(2\pi\)
\(0\)
\(\lim_{x\rightarrow 0}\left ( \frac{\int_{0}^{x^2}\,sin\,\sqrt{t}\,dt}{x^2} \right )=\)
Please choose your answer from the right side options
\(\frac{2}{3}\)
\(\frac{2}{9}\)
\(\frac{1}{3}\)
\(0\)
\(\frac{1}{6}\)
The area bounded by \(y=sin^2x,x=\frac{\pi }{2} \) and \(x=\pi \) is
Please choose your answer from the right side options
\(\frac{\pi}{2}\)
\(\frac{\pi}{4}\)
\(\frac{\pi}{8}\)
\(\frac{\pi}{16}\)
\(2\pi\)
If ω represents a complex cube root of unity, then \(\Large \overset{9}{\underset{r=1} \Sigma} r(r+1- \omega)(r +1 -\omega^2 ) =\)
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5025
4020
2016
3015
If α and β are the roots of \(\large x^2 +7x +3 = 0\) and \(\Large \frac{2\alpha}{3-4\alpha} , \frac{2\beta}{3-4\beta}\) are the roots of \(\large ax^2 + bx +c=0\) and GCD of a,b,c is 1 then a +b + c =
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11
0
243
81
If α, β are the roots of \(\large x^2 +bx+c=0\), γ , δ are the roots of \(\large x^2 + b_1 x + c_1 = 0\) and \(\large \gamma < \alpha < \delta < \beta \) then \(\large (c-c_1)^2 <\)
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\(\large (b_1 -b)(bc_1-b_1c)\)
1
\(\large (b -b_1)^2\)
\(\large (c -c_1)(b_1c-b_1c_1)\)
Let a,b,c be the sides of a scalene triangle. If λ is a real number such that root of the equation \(\large x^2 + 2(a+b+c)x + 3\lambda(ab+bc+ca)=0\) are real then the interval in which λ lies is
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\(\Large (- \infty, \frac{4}{3})\)
\(\Large ( \frac{5}{3}, \infty )\)
\(\Large (\frac{1}{3}, \frac{5}{3})\)
\(\Large (\frac{4}{3}, \infty)\)
The polynomial equation of degree 4 having real coefficients with three of its roots as \(\large 2 \ \pm \sqrt 3\) and 1+ 2 i, is
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\(\large x^4 - 6x^3 - 14x^2 +22x +5=0\)
\(\large x^4 - 6x^3 - 19x+22x -5=0\)
\(\large x^4 - 6x^3 + 19x - 22x +5=0\)
\(\large x^4 - 6x^3 + 14x^2 -22x +5=0\)
All letters of the word ANIMAL are permitted in all possible ways and the permutations thus formed are arranged in dictionary order. If the rank of word ANIMAL is x. Then the permutation with rank x , among the permutation obtained by permuting the letter of the word PERSON and arranging the permutations thus formed in the dictionary order is
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ENOPRS
NOSPRE
NOEPRS
ESORNP
A student is allowed to choose atmost n books from a collection of 2n + 1 books. If the total number of ways in which he can select atleast one book is 255, then the value of n is
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4
5
6
7
The sum of all the coefficients in the binomial expansion of \(\large (1+2x)^n\) is 6561. Let R = \(\large (1+2x)^n\) = I + F where I ∈ N and 0 < F < 1 . If \(\Large x= \frac{1}{\sqrt 2}\) , then \(\Large 1 - \frac{F}{1 + (\sqrt 2 - 1) ^4} =\)
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\(\large (3\sqrt 2 -4)\)
\(\large 4(3\sqrt 2 +4)\)
\(\large (\sqrt 2 -1)^4\)
1
If \(\Large \frac{(1-px)^{-1}}{1-qx} = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ...,\) then \(\large a_n =\)
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\(\Large \frac{p^{n+1} -\ q^{n+1}}{q \ - \ p}\)
\(\Large \frac{p^{n+1} -\ q^{n+1}}{p \ - \ q}\)
\(\Large \frac{p^n -\ q^n}{q \ - \ p}\)
\(\Large \frac{p^n -\ q^n}{p \ - \ q}\)
If \(\Large \frac{3}{(x-1)(x^2+x+1)} = \frac{1}{x-1} - \frac{x+2}{(x^2+x+1)} = f_1(x) - f_2(x)\) and \(\Large \frac{x+1}{(x-1)^2(x^2+x+1)} = Af_1(x) + ( B + \frac{D}{x-1}) f_2(x) + \frac{C}{(x-1)^2}\) then A + B + C + D =
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1
\(\Large - \frac{1}{3}\)
0
\(\Large \frac{1}{3}\)
Let M and m respectively denote the maximum and minimum values of \(\large [f(\theta)]^2\) where \(\large f(\theta) = \sqrt{a^2 cos^2 \theta + b^2 sin^2 \theta } + \sqrt{a^2 sin^2 \theta + b^2 cos^2 \theta }\). Then M - m =
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\(\large a^2 + b^2\)
\(\large (a - b)^2\)
\(\large a^2 b^2\)
\(\large (a + b)^2\)
If cos A = \(\Large - \frac{60}{61}\)and tan B = \(\Large - \frac{7}{24}\) and neither A nor B is in the second quadrant, then the angle \(\Large A + \frac{B}{2}\) lies in the quadrant
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1
2
3
4
\(\large cos^2\) 5° - \(\large cos^2\) 15° - \(\large sin^2 \) 15° + \(\large sin^2 \)35° + cos 15° sin 15° - cos 5° sin 35° =
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0
1
\(\Large \frac{3}{2}\)
2
If cos θ ≠ 0 and sec θ -1 =( \(\large \sqrt 2\) - 1) tan θ then θ =
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\(\Large n \pi + \frac{\pi}{8}, n \ \epsilon \ Z\)
\(\Large 2n \pi + \frac{\pi}{4} \ (or)\ 2n\pi, n \ \epsilon \ Z\)
\(\Large 2n \pi + \frac{\pi}{8}, n \ \epsilon \ Z\)
\(\Large 2n \pi - \frac{\pi}{4} (or) \ 2n\pi, n \ \epsilon \ Z\)
\(\large cot [\overset{32} {\underset{n=3} \Sigma} cot^{-1}(1 + \overset{n} {\underset{k=1} \Sigma} 2k)] =\)
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\(\Large \frac{10}{3}\)
\(\Large \frac{8}{3}\)
\(\Large \frac{14}{3}\)
\(\Large \frac{16}{3}\)
If sin x cos hy = cosθ, cos x sin hy = sinθ and 4 tan x = 3. Then \(\large sin \ h^2y\) =
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\(\Large \frac{4}{5}\)
\(\Large \frac{9}{16}\)
\(\Large \frac{9}{25}\)
\(\Large \frac{16}{25}\)
In triangle ABC, if \(\Large \frac{b+c}{9} = \frac{c+a}{10} = \frac{a+b}{11} = k \) then \(\Large \frac{cos \ A \ + \ cos \ B}{cos \ C} =\)
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\(\Large \frac{9}{10}\)
\(\Large \frac{10}{11}\)
\(\Large \frac{11}{12}\)
\(\Large \frac{12}{13}\)
In triangle ABC, with usual notation, match the items in list I with the items in list II and choose the correct option
List I List II
A) \(\Large r_1 r_2 \sqrt{(\frac{4R -r_1 -r_2}{r_1 + r_2})}\) I) b
B) \(\Large \frac{r_2 (r_3 + r_2)}{\sqrt{r_1r_2 + r_2r_3 + r_3r_1}}\) II) \(\large a^2 , b^2 , c^2\) are in A.P
C) \(\Large \frac{a}{c} = \frac{sin \ (A-B)}{sin \ (B-C)}\) III) \(\large \Delta\)
D) \(\large bc \ cos^2 \frac{A}{2}\) IV) \(\large Rr_1r_2r_3\)
V) s(s-a)
The correct match is
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A B C D
IV III I V
A B C D
V IV III II
A B C D
III I II V
A B C D
IV V II I
If a ,b , c are the sides of ∆ABC for which \(\large r_1=8, \ r_2=12\) and \(\large r_3=24\) then the ordered triad (a, b,c) =
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(12,20,16)
(12,16,20)
(16,12,20)
(20,16,12)
If \(\large 4 \bar i+ 7 \bar j + 8 \bar k, \ 2 \bar i+ 3 \bar j + 4 \bar k, \ 2 \bar i+ 5 \bar j + 7 \bar k\) are respectively the positions vectors of vertices A,B,C of triangle ABC, then the position vectors of the point where the bisector of angle meets \(\large \overline {BC}\) is
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\(\Large 2 \bar i + \frac{13}{3} \bar j + 2 \bar k\)
\(\Large 2 \bar i - \frac{13}{3} \bar j + 6 \bar k\)
\(\Large 2 \bar i + 13 \bar j + 6 \bar k\)
\(\Large 2 \bar i + \frac{13}{3} \bar j + 6 \bar k\)
The equation of the plane passing through the points \(\large \bar i + 2 \bar j - \bar k\) and perpendicular to the line of intersection of the plane \(\large \bar r . ( 3\bar i - \bar j + \bar k) = 1\) and \(\large \bar r . ( \bar i + 4 \bar j -2 \bar k) = 2\) is
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\(\large \bar r . ( -2\bar i - 5\bar j + \bar k) = 0\)
\(\large \bar r . ( \bar i + 7 \bar j + 4\bar k) = 0\)
\(\large \bar r . ( 2\bar i - 7\bar j -13 \bar k) = 1\)
\(\large \bar r . ( -2\bar i +7 \bar j + 13\bar k) = 0\)
If the position vector of the vertices, A, B and C of \(\large \Delta ABC\) are \(\large \bar i + 2 \bar j - 5 \bar k, -2 \bar i + 2 \bar j + \bar k \) and \(\large 2 \bar i + \bar j - \bar k\) respectively then ∠B =
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\(\Large cos^{-1} (\frac{7}{3\sqrt{10}})\)
\(\Large cos^{-1} (\frac{8}{\sqrt{105}})\)
\(\Large cos^{-1} (\frac{1}{\sqrt{42}})\)
\(\Large cos^{-1} (-\frac{7}{3\sqrt{10}})\)
If the position vector of the vertices of \(\large \Delta ABC\) are \(\large \overline {OA} = 3 \bar i + \bar j + 2 \bar k ,\ \overline {OB} = \bar i + 2 \bar j + 3 \bar k\) and \(\large \overline {OC} = 2 \bar i + 3 \bar j + \bar k\) then the length of altitude of triangle ABC drawn from A is
Please choose your answer from the right side options
\(\Large \sqrt{\frac{3}{2}}\)
\(\Large \frac{3}{\sqrt 2}\)
\(\Large \frac{\sqrt 3}{2}\)
\(\Large \frac{3}{2}\)
A new tetrahedron is formed by joining the faces of a given tetrahedron OABC. Then the ratio of volume of new tetrahedron to that of given tetrahedron is
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\(\Large \frac{3}{25}\)
\(\Large \frac{1}{27}\)
\(\Large \frac{5}{62}\)
\(\Large \frac{1}{162}\)
Let \(\large \bar A = 2 \bar i + \bar j - 2 \bar k\) and \(\large \bar B = \bar i + \bar j .\, if \, \bar C\) is a vector such that \(\large \bar A\ . \overline C = |\overline C|, |\overline C - \bar A| = 2 \sqrt 2\) and the angle between \(\large \bar A \times \overline B\) and \(\large \overline C\) is 30° then the value of \(\large |(\bar A \times \overline B)\times \overline C|\) is
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\(\Large \frac{2}{3}\)
\(\Large \frac{3}{2}\)
3
2
If \(\Large a_0, a_1, ...a_{11}\) are in an arithmetic progression with common difference d , then their mean deviation from arithmetic mean is
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\(\Large \frac{30}{11} |d|\)
2|d|
3|d|
12 |d|
The variance of the following continuous frequency distribution is
| Class Interval | 0 - 10 | 10 - 20 | 20 - 30 | 30-40 |
| Frequency | 2 | 3 | 4 | 1 |
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201
62
19
84
If two sections of strength 30 and 45 are formed from 75 students who are admitted in a school, then the probability that two particular students are always together in the same section is
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\(\Large \frac{66}{185}\)
\(\Large \frac{19}{37}\)
\(\Large \frac{29}{185}\)
\(\Large \frac{18}{37}\)
A bag contains 2n coins out of which n-1 are unfair with head on both sides and remaining are fair. One coin is picked from a bag at random and tossed. If the probability that head falls in the toss is \(\Large \frac{41}{56}\) then the number of unfair coins in the bag is
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18
15
13
14
Bag A contains 6 Green and 8 Red balls and bag B contains 9 Green and 5 Red balls. A card is drawn at random from a well shuffled pack of 52 playing cards. If it is a spade, two balls are drawn at random from bag A, otherwise two balls are drawn at random from bag B. If the two balls drawn are found to be of the same colour, then the probability that they are drawn from bag A is
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\(\Large \frac{43}{181}\)
\(\Large \frac{1}{4}\)
\(\Large \frac{48}{131}\)
\(\Large \frac{43}{138}\)
A random variable X has the probability distribution,
| X = \(\large x_i\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| P (X = \(\large x_i\) ) | 0.2 | 0.3 | 0.12 | 0.1 | 0.2 | 0.08 |
If A = {\(\large x_i\) /\(\large x_i\) is a prime number}, B ={ \(\large x_i\) /\(\large x_i\) < 4} are two events then P (A \(\large \cup\) B) =
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0.31
0.62
0.82
0.41
In a Poisson distribution with mean \(\large \overset{\infty}{\underset{x=0} \Sigma} |x-\bar{x}| P(X=x) =\)
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e
\(\Large \frac{1}{e}\)
\(\Large \frac{2}{e}\)
\(\Large \frac{2}{3e}\)
Two straight rods of length 2a and 2b move along the coordinate axis in such a way that their extremities are always concyclic. Then the locus of center of such circles is
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\(\large 2(x^2 + y^2) = a^2 + b^2\)
\(\large 2(x^2 - y^2) = a^2 + b^2\)
\(\large x^2 + y^2 = a^2 + b^2\)
\(\large x^2 - y^2 = a^2 - b^2\)
When the coordinate axes are rotated around the origin in the positive direction through an
angle \(\Large \frac{\pi}{4}\), if the equation \(\large 25x^2 + 9 y^2 = 225\) is transformed to \(\large \alpha x^2 + \beta xy + \gamma y^2 = \delta\) the \(\large (\alpha + \beta + \gamma - \sqrt{\delta})^2 =\)
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3
9
4
16
The equation of the line through the point of intersection of the lines 3x -4y + 1 = 0 and 5x + y -1 = 0 and making equal non-zero intercepts on the coordinate axes is
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2x + 2y = 3
23x + 23y = 6
23x + 23y = 11
2x + 2y = 7
The line through P(a,2) where \(\large a \neq 0\), making an angle 45° with the positive direction of the X-axis meets the curve \(\Large \frac{x^2}{9} + \frac{y^2}{4} = 1\) at A and D and the coordinate axis at B and C. If PA, PB, PC and PD are in geometric progression then 2a =
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13
7
1
-13
The equation of the perpendicular bisector of the sides AB and AC of a ΔABC are x - y = 5 and x+ 2y = 0 respectively. If A is (1, -2 ) then the equation of straight line BC is
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14x + 23y -40 = 0
12x + 17y - 28 = 0
14x - 29y -30 = 0
7x - 12y + 15 = 0
If each line of a pair of lines original line passing through origin is at a perpendicular distance of 4 units from the point ( 3, 4 ), then the equation of the pair of lines is
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\(\large 7x^2 + 24xy =0\)
\(\large 7y^2 + 24xy =0\)
\(\large 7y^2 - 24xy =0\)
\(\large 7x^2 - 24xy =0\)
Variable straight lines y = mx + c make intercepts on the curve \(\large y^2 - 4ax = 0\) which subtend a right angle at the origin. Then the point of concurrence of these lines y = mx + c is
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(4a, 0)
(2a, 0)
(-4a, 0)
(-2a, 0)
The abscissae of two points P, Q are the roots of the equation \(\large 2x^2 + 4x - 7= 0\) and their ordinates are the roots of the equation \(\large 3x^2 - 12x - 1= 0\) . Then the centre of the circle with PQ as a diameter is
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(-1, 2)
(-2, 6)
(1, -2)
(2, -6)
If the angle between a pair of tangents drawn from a point P to the circle \(\large x^2 + y^2 + 4x -6y + 9sin^2 \alpha + 13cos^2 \alpha = 0\) is \(\large 2 \alpha\), then the equation of the locus of P is
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\(\large x^2 + y^2 + 4x -6y + 4 =0\)
\(\large x^2 + y^2 + 4x -6y - 9 =0\)
\(\large x^2 + y^2 - 4x + 6y - 4 =0\)
\(\large x^2 + y^2 + 4x -6y + 9 =0\)
The equation of the circle whose radius is 3 and which touches internally the circle \(\large x^2 + y^2 - 4x - 6y -12 =0\) at the point (-1, -1 ) is
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\(\large 5x^2 + 5y^2 +9x -6y-7=0\)
\(\large 5x^2 + 5y^2 -8x -14y-32=0\)
\(\large 5x^2 + 5y^2 -6x +8y-8=0\)
\(\large 5x^2 + 5y^2 +6x -8y-12=0\)
Suppose that the circle \(\large x^2 + y^2 +2gx +2fy +c =0\) has its centre on 2x + 3y - 7 =0 and cut the circles \(\large x^2 + y^2 -4x -6y +11 =0\) and \(\large x^2 + y^2 -10 x -4y + 21 =0\) orthogonally. Then 5g -10f +3c =
Please choose your answer from the right side options
0
1
3
9
If the radical axis of the circle \(\large x^2 + y^2 + 2gx+ 2fy +c=0\) and \(\large 2x^2 + 2y^2 + 3x+ 8y +2c=0\) touches the circle \(\large x^2 + y^2 + 2x+ 2y +1=0\) then (4g - 3)(f - 2) =
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0
-1
1
2
The parabola \(\large x^2 = 4ay\) makes an intercept of length \(\large \sqrt{40}\) units on the line y = 1 +2x then a value of 4a is
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2
-2
-1
2
The locus of the point of intersection of perpendicular normals to the parabola \(\large y^2 =4 ax\) is
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\(\large y^2 -2ax + a^2=0\)
\(\large y^2 +ax + 2a^2=0\)
\(\large y^2 -ax + 2 a^2=0\)
\(\large y^2 -ax +3 a^2=0\)
P is a variable point on the ellipse \(\Large \frac{x^2}{a^2} + \frac{y^2}{b^2}= 1\) with foci \(\large F_1\) and \(\large F_2\). If A is the area of the triangle \(\large PF_1F_2\), then the maximum value of A is
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\(\Large \frac{e}{ab}\)
\(\Large \frac{ae}{b}\)
\(\Large aeb\)
\(\Large \frac{ab}{e}\)
If the line joining the points A (α) and B (β) on the ellipse \(\Large \frac{x^2}{25} + \frac{y^2}{9} = 1\) is a focal chord, then one possible value of \(\Large cot\frac{\alpha}{2} cot \frac{\beta}{2}\) is
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-3
3
-9
9
The equation of a tangent to the hyperbola \(\large 16x^2 - 25y^2 -96x + 100y - 356 = 0\) which makes an angle 45° with its transverse axis is
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x - y + 2 = 0
x - y + 4 = 0
x + y + 2 = 0
x + y + 4 = 0
If P (0,7,10), Q(-1, 6, 6) and R(-4, 9, 6) are three points in the space, then PQR is
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Right angled isosceles triangle
Equilateral triangle
Isosceles but not right angled triangle
Scalene triangle
A(2, 3, 5), B (\(\large \alpha\), 3, 3) and C(7, 5, \(\large \beta\)) are the vertices of a triangle. If the median through A is equally inclined with the coordinate axes then \(\Large Cos^{-1} (\frac{\alpha}{\beta})\) =
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\(\Large Cos^{-1} (\frac{-1}{9})\)
\(\Large \frac{\pi}{2}\)
\(\Large \frac{\pi}{3}\)
\(\Large Cos^{-1} (\frac{2}{5})\)
The plane 3x+ 4y + 6z+ 7 = 0 is rotated about the line \(\large \bar r = (\bar i + 2 \bar j - 3 \bar k) + t(2\bar i - 3 \bar j + \bar k)\) until the plane passes through the origin. The equation of the plane in new position is
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x + y + z = 0
6x + 3y - 4z = 0
4x - 5y - 2z = 0
x + 2y + 4z = 0
If \(\Large \underset{x \rightarrow \infty} {lim} \{ \frac{x^3 + 1}{x^2 + 1} - (\alpha x + \beta) \}\) exist and equal to 2 then the ordered pair (α , β) of real numbers is
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(1, -1)
(-2, 1)
(-1, 1)
(1, -2)
For k>0, \(\Large \overset{\infty}{\underset{x=0} \Sigma} \frac{k^x}{x!} \underset{x \rightarrow \infty} {lim} \frac{n!}{(n-x)!} (1 - \frac{k}{n})^{n-x} (\frac{1}{n})^x = \)
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0
k
x
1
Let \(\large f: \mathbb{R} \rightarrow \mathbb{R}\) be the function defined by
\(\large f(x) = \left [ \begin{array}{ll} 5 & \mbox{if } x \leq 1 \\ a +bx & \mbox{if } 1< x < 3 \\ b +5x & \mbox{if } 3 \leq x <5 \\ 30 & \mbox{if } x \geq 5 \end{array} \right.\) then f is
then \(\large f\) is
Please choose your answer from the right side options
continuous if a = 5 and b = 5
continuous if a = 0 , b = 5
continuous if a = − 5, b =10
not continuous for any value of a and b
Let [ x ] denote the greatest integer less than or equal to x. Then the number of points where the function y = [ x ] + |1 - x| , \(\large 1 \leq x \leq 3\) is not differentiable, is
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1
2
3
4
If \(\large \sqrt{1 - x^6} + \sqrt{1-y^6} = a(x^3 - y^3)\) , then \(\Large y^2 \frac{dy}{dx} =\)
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\(\Large \sqrt{\frac{1-y^6}{1-x^6}}\)
\(\Large x \sqrt{\frac{1-y^6}{1-x^6}}\)
\(\Large x^2 \sqrt{\frac{1-y^6}{1-x^6}}\)
\(\Large \frac{1}{x^2} \sqrt{\frac{1-y^6}{1-x^6}}\)
If y = f(x) is twice differentiable function such that a point P, \(\Large \frac{dy}{dx} =4, \frac{d^2y}{dx^2} = -3\), then \(\Large ( \frac{d^2x}{dy^2})_P = \)
Please choose your answer from the right side options
\(\Large \frac{64}{3}\)
\(\Large \frac{16}{3}\)
\(\Large \frac{3}{16}\)
\(\Large \frac{3}{64}\)
The time T of oscillation of a simple pendulum of length L is governed by, \(\Large T =2\pi \sqrt{\frac{L}{g}}\)where g is constant. The percentage by which length be changed in order to correct an error of loss equal to 2 minutes of time per day is
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\(\Large - \frac{5}{18}\)
\(\Large - \frac{2}{9}\)
\(\Large \frac{1}{6}\)
\(\Large \frac{1}{9}\)
Let A, G, H and S respectively denote the arithmetic mean, geometric mean, harmonic mean and the sum of the numbers \(\large a_1, a_2, a_3,...,a_n\). Then the value of x at which the function \(\large f(x) = \overset{n}{\underset{k=1} \Sigma} (x - a_k)^2\) has minimum is
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S
H
G
A
For m > 1 , n > 1 , the value of c for which the Rolle’s theorem is applicable for the function \(\large f(x)=x^{2m-1} (a- x)^{2n}\) in (0, a) is
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\(\Large \frac{2am-1}{m+2n-1}\)
\(\Large \frac{a(m-n+1)}{2m+2n}\)
\(\Large \frac{a(2m-1)}{2m+2n-1}\)
\(\Large \frac{a(2m+1)}{m+n-1}\)
If the function \(\large f : [-1,1] \rightarrow \mathbb{R}\) defined by
\(\large f(x) = \left\{ \begin{array}{ll} 2^x + 1 & \mbox{if } x \ \epsilon \ [-1,0) \\ 1 & \mbox{if } x = 0 \\ 2^x - 1 & \mbox{if } x \ \epsilon \ (0,1] \end{array} \right.\)then in [−1, 1] , f ( x ) has
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a maximum
a minimum
both maximum and minimum
neither maximum nor minimum
\(\Large \int \frac{x-1}{(x+1)\sqrt{x^3 + x^2 +x}} dx=\)
Please choose your answer from the right side options
\(\Large 2 \ Tan^{-1} (\sqrt{\frac{1+x+x^2}{x}}) + c\)
\(\Large \ Tan^{-1} (\sqrt{\frac{1+x+x^2}{x}}) + c\)
\(\Large 2 \ Tan^{-1} (\sqrt{\frac{x}{1+x+x^2}}) + c\)
\(\Large \ Tan^{-1} (\sqrt{\frac{1+x^2}{x}}) + c\)
If \(\large I(x)= \int x^2 (log \ x)^2 dx\) and I (1) = 0 , then I (x) =
Please choose your answer from the right side options
\(\Large \frac{x^3}{18}[8 (log \ x)^2 - 3 \ log \ x] + \frac{7}{18}\)
\(\Large \frac{x^3}{27}[9 (log \ x)^2 + 6 \ log \ x] - \frac{2}{27}\)
\(\Large \frac{x^3}{27}[9 (log \ x)^2 - 6 \ log \ x + 2] - \frac{2}{27}\)
\(\Large \frac{x^3}{27}[9 (log \ x)^2 - 6 \ log \ x - 2] + \frac{2}{27}\)
\(\Large \int \frac{x^5 dx}{(x^2 + x +1)(x^6 + 1)(x^4 - x^3 + x -1)} =\)
Please choose your answer from the right side options
\(\Large log_e |\frac{x^6 - 1}{x^6 +1}|+ c\)
\(\Large \frac{1}{12}log_e |\frac{x^6 - 1}{x^6 +1}|+ c\)
\(\Large \frac{1}{12} log_e |\frac{x^4 + 1}{x^4 -1}|+ c\)
\(\Large log_e |\frac{x^8 + 4}{x^6 -1}|+ c\)
\(\Large \int \frac{dx}{x + \sqrt{x-1}} =\)
Please choose your answer from the right side options
\(\Large log_e|x + \sqrt{x-1}| - \frac{1}{\sqrt 3}Tan^{-1} (\frac{2\sqrt{x-1} + 1}{\sqrt 3}) + c\)
\(\Large \frac{1}{\sqrt 3}log_e|x + \sqrt{x-1}| - Tan^{-1} (\frac{2\sqrt{x-1} + 1}{\sqrt 3}) + c\)
\(\Large \frac{2}{\sqrt 3}log_e|x + \sqrt{x-1}| - Tan^{-1} (\frac{2\sqrt{x-1} + 1}{\sqrt 3}) + c\)
\(\Large log_e|x + \sqrt{x-1}| - \frac{2}{\sqrt 3}Tan^{-1} (\frac{2\sqrt{x-1} + 1}{\sqrt 3}) + c\)
\(\Large \overset{x}{\underset{log_e 2} \int}\frac{dt}{\sqrt{e^t - 1}} = \frac{\pi}{6} \implies x=\)
Please choose your answer from the right side options
\(\large 2 . log_e 2\)
\(\large 3 . log_e 2\)
\(\large 4 . log_e 2\)
\(\large 8 . log_e 2\)
\(\Large \overset 1{\underset 0 \int} \frac{log_e (1+x)}{1+x^2} dx =\)
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\(\Large \frac{\pi}{4} log_e 2\)
\(\Large \frac{\pi}{6} log_e 6\)
\(\Large \frac{\pi}{2} log_e 8\)
\(\Large \frac{\pi}{8} log_e 2\)
If the area of the circle \(\large x^2 + y^2 = 2\) is divided into parts by the parabola \(\large y = x^2\), then the area (in sq. units) of the larger part is
Please choose your answer from the right side options
\(\Large \frac{3\pi}{2} - \frac{1}{3}\)
\(\Large 6\pi - \frac{4}{3}\)
\(\Large \frac{4\pi}{3} - \frac{2}{3}\)
\(\Large 4\pi - \frac{1}{4}\)
If c is a parameter, then the differential equation of the family of curves \(\large x^2 = c(y+c)^2\) is
Please choose your answer from the right side options
\(\Large x(\frac{dy}{dx})^3 + y(\frac{dy}{dx})^2 - 1 = 0\)
\(\Large x(\frac{dy}{dx})^3 - y(\frac{dy}{dx})^2 + 1 = 0\)
\(\Large x(\frac{dy}{dx})^3 + y(\frac{dy}{dx})^2 + 1 = 0\)
\(\Large x(\frac{dy}{dx})^3 - y(\frac{dy}{dx})^2 - 1 = 0\)
If \(\large f(x), f'(x), f''(x)\) are positive functions and \(\large f(0)=1, f'(0)=2\), then the solution of the differential equation \(\large \left | {\begin{array}{cc} f(x) & f'(x) \\ f'(x) & f''(x) \\ \end{array} } \right | = 0 \) is
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\(\large e^{2x}\)
\(\large 2 \ sin \ x + 1\)
\(\large sin^2 x + 2x + 1\)
\(\large e^{4x}\)
A bag contains 5 red balls, 3 black balls and 4 white balls are drawn at random. The probability that they are not of same colour is
Please choose your answer from the right side options
\(\frac{37}{44}\)
\(\frac{31}{44}\)
\(\frac{21}{44}\)
\(\frac{41}{44}\)
The radical centre of the circles
\(x^2+y^2-4x-6y+5=0,x^2+y^2-2x-4y-1=0,x^2+y^2-6x-2y=0 \) lies on the line
Please choose your answer from the right side options
\(x+y-5=0 \)
\(2x-4y+7=0 \)
\(4x-6y+5=0 \)
\(18x-12y+1=0\)
If \(cosec\theta -cot\theta =2017\) , then quadrant in which \(\theta\) lies is
Please choose your answer from the right side options
I
IV
III
II
If \(\int e^{2x}f'(x)dx=g(x) \) , then \(\int (e^{2x}f(x)+e^{2x}f'(x))dx= \)
Please choose your answer from the right side options
\(\frac{1}{2}[e^{2x}f(x)-g(x)]+C \)
\(\frac{1}{2}[e^{2x}f(x)+g(x)]+C \)
\(\frac{1}{2}[e^{2x}f(2x)+g(x)]+C \)
'>\(\frac{1}{2}[e^{2x}f'(2x)+g(x)]+C\)
If \(A=(5,3),B=(3,-2)\) and a point P is such that the area of the triangle PAB is 9, then the locus of P represents
Please choose your answer from the right side options
a circle
a pair of coincident lines
a pair of parallel lines
a pair of perpendicular lines