Math Practice MCQs

Mathematics - Set Theory

1. Let $S = \{2, 4, 6, 8, \dots, 20\}$. What are the maximum number of subsets of $S$?

A.10B.20C.512D.1024

Mathematics - Relations and Functions

2. Suppose $X = \{1, 2, 3, 4\}$ and $R$ is a relation on $X$. If $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)\}$, then which one of the following is correct?

A.$R$ is reflexive and symmetric but not transitiveB.$R$ is symmetric and transitive but not reflexiveC.$R$ is reflexive and transitive but not symmetricD.$R$ is neither reflexive nor transitive but symmetric

Mathematics - Quadratic Equations

3. If the sum of the roots of the equation $ax^2 + bx + c = 0$ is equal to the sum of their squares, then

A.$a^2 + b^2 = c^2$B.$a^2 + b^2 = a + b$C.$ab + b^2 = 2ac$D.$ab - b^2 = 2ac$

Mathematics - Sequences and Series

4. The four arithmetic means between $6$ and $-9$ are $a_1, a_2, a_3, a_4$. What is the value of $a_1 + a_3$?

A.-1B.0C.6D.-3

Mathematics - Binomial Theorem

5. The number of terms in the expansion of $(x + a)^{100} + (x - a)^{100}$ after simplification is

A.202B.101C.51D.50

Mathematics - Matrices and Determinants

6. If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, then the value of $A^4$ is

A.$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$B.$\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$C.$\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}$D.$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Mathematics - Matrices and Determinants

7. The system of equations $kx + y + z = 1$, $x + ky + z = k$ and $x + y + kz = k^2$ has no solution if $k$ equals

A.0B.1C.-1D.-2

Mathematics - Permutations and Combinations

8. How many different ways are there to select 11 cricket players out of 20 players?

A.${}^{20}P_{11} \times 11!$B.${}^{20}P_{11}$C.${}^{20}C_{11} \times 11!$D.${}^{20}C_{11}$

Mathematics - Complex Numbers

9. If $\left[(1 - i)^2 \cdot i^5 + (1 + i)^2\right]^{-1} = x + iy$, then what is the value of $x + y$?

A.0B.1C.1/2D.-1/2

Mathematics - Vectors

10. If the magnitude of difference of two unit vectors is $\sqrt{3}$, then the magnitude of sum of the two vectors is

A.$\frac{1}{2} \text{ unit}$B.$1 \text{ unit}$C.$2 \text{ units}$D.$3 \text{ units}$

Mathematics - 3D Geometry

11. Let $A(2, -4, 5)$ and $B(-2, 1, 2)$. What are the direction cosines of the line parallel to the line joining points $A$ and $B$?

A.$\left(-\frac{4}{\sqrt{50}}, \frac{5}{\sqrt{50}}, -\frac{3}{\sqrt{50}}\right)$B.$\left(\frac{4}{\sqrt{50}}, -\frac{5}{\sqrt{50}}, \frac{3}{\sqrt{50}}\right)$C.$\left(\frac{2}{\sqrt{30}}, \frac{2}{\sqrt{30}}, \frac{2}{\sqrt{30}}\right)$D.$\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$

Mathematics - Trigonometry

12. If $\sin(A+B) = 1$ and $\cos(A-B) = \frac{\sqrt{3}}{2}$, then find the angle of $A$ and $B$:

A.$A = 45^\circ, B = 45^\circ$B.$A = 30^\circ, B = 60^\circ$C.$A = 75^\circ, B = 15^\circ$D.$A = 60^\circ, B = 30^\circ$

Mathematics - Trigonometry

13. If $\sin x + \sin y + \sin z = 3$, then $\cos 4x + \cos 4y + \cos 4z$ is

A.3B.2C.1D.0

Mathematics - Inverse Trigonometry

14. The value of $\sin^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right)$ is equal to

A.0B.$\frac{\pi}{4}$C.$\frac{\pi}{3}$D.$\frac{\pi}{2}$

Mathematics - Trigonometry

15. If the angles of a triangle $ABC$ are in the ratio $1 : 2 : 3$, then the corresponding sides are in the ratio

A.$1 : 2 : 3$B.$3 : 2 : 1$C.$1 : \sqrt{3} : 2$D.$1 : \sqrt{3} : \sqrt{2}$

Mathematics - Coordinate Geometry

16. A straight line cuts off an intercept of 2 units on the positive direction of X-axis and passes through the point $(-3, 5)$. What is the foot of the perpendicular drawn from the point $(3, 3)$ on this line?

A.$(1, 3)$B.$(2, 0)$C.$(0, 2)$D.$(1, 1)$

Mathematics - Circles

17. The equation $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$ represents a circle whose centre is

A.$\left(\frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2}\right)$B.$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$C.$(x_1, y_1)$D.$(x_2, y_2)$

Mathematics - Parabola

18. If $t_1$ and $t_2$ be the parameters of the end points of a focal chord for the parabola $y^2 = 4ax$, then which one is true?

A.$t_1 t_2 = 1$B.$\frac{t_1}{t_2} = 1$C.$t_1 t_2 = -1$D.$t_1 + t_2 = -1$

Mathematics - Ellipse

19. The latus rectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is

A.$\frac{1}{\sqrt{6}}$B.$\frac{\sqrt{3}}{2}$C.$\frac{\sqrt{3}}{4}$D.1/2

Mathematics - Conic Sections

20. If $t$ is a parameter, then $x = a\left(t + \frac{1}{t}\right), y = b\left(t - \frac{1}{t}\right)$ represents

A.an ellipseB.a circleC.a pair of straight linesD.a hyperbola

Mathematics - Calculus

21. The $\lim_{x \to 2} \frac{5}{\sqrt{2} - \sqrt{x}}$ is

A.$10\sqrt{2}$B.$+\infty$C.$-\infty$D.does not exist

Mathematics - Calculus

22. For the function $f(x) = e^{\cos x}$, Rolle's theorem is

A.applicable when $\frac{\pi}{2} \le x \le \frac{3\pi}{2}$B.applicable when $0 \le x \le \frac{\pi}{2}$C.applicable when $0 \le x \le \pi$D.applicable when $\frac{\pi}{4} \le x \le \frac{\pi}{2}$

Mathematics - Calculus

23. The function $f(x) = e^{ax} + e^{-ax}$, $a > 0$ is monotonically increasing for

A.$-1 < x < 1$B.$x < -1$C.$x > -1$D.$x > 0$

Mathematics - Calculus

24. Let $f(x) = \tan^{-1}x$. Then $f'(x) + f''(x) = 0$, when $x$ is equal to

A.0B.1C.$i$D.$-i$

Mathematics - Application of Derivatives

25. If the rate of increase of the radius of a circle is $5\text{ cm/sec}$, then the rate of increase of its area when the radius is $20\text{ cm}$ will be

A.$10\pi$B.$20\pi$C.$200\pi$D.$400\pi$

Mathematics - Integral Calculus

26. The value of the integral $\int \frac{dx}{(e^x + e^{-x})^2}$ is

A.$\frac{1}{2}(e^{2x} + 1) + C$B.$\frac{1}{2}(e^{-2x} + 1) + C$C.$-\frac{1}{2}(e^{2x} + 1)^{-1} + C$D.$\frac{1}{4}(e^{2x} - 1) + C$

Mathematics - Integral Calculus

27. What is the area bounded between the parabola $y^2 = x$ and the vertical line $x = 1$, in the first quadrant?

A.2/3B.1/2C.1/3D.1

Mathematics - Definite Integrals

28. $\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} \, dx =$

A.$\frac{3\pi}{2}$B.$\frac{\pi}{2}$C.$\frac{\pi}{3}$D.$2\pi$

Mathematics - Probability

29. For two events $A$ and $B$, it is given that $P(A) = 3/5$, $P(B) = 3/10$ and $P(A/B) = 2/3$. If $\bar{A}$ and $\bar{B}$ are the complementary events of $A$ and $B$, then what is $P(\bar{A}/\bar{B})$ equal to?

A.3/7B.3/4C.1/3D.4/7

Mathematics - Statistics

30. If the covariance between $x$ and $y$ is 30, variance of $x$ is 25 and variance of $y$ is 144, then what is the correlation coefficient?

A.0.4B.0.5C.0.6D.0.7

1. Let $S = \{2, 4, 6, 8, \dots, 20\}$. What are the maximum number of subsets of $S$?

2. Suppose $X = \{1, 2, 3, 4\}$ and $R$ is a relation on $X$. If $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)\}$, then which one of the following is correct?

3. If the sum of the roots of the equation $ax^2 + bx + c = 0$ is equal to the sum of their squares, then

4. The four arithmetic means between $6$ and $-9$ are $a_1, a_2, a_3, a_4$. What is the value of $a_1 + a_3$?

5. The number of terms in the expansion of $(x + a)^{100} + (x - a)^{100}$ after simplification is

6. If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, then the value of $A^4$ is

7. The system of equations $kx + y + z = 1$, $x + ky + z = k$ and $x + y + kz = k^2$ has no solution if $k$ equals

8. How many different ways are there to select 11 cricket players out of 20 players?

9. If $\left[(1 - i)^2 \cdot i^5 + (1 + i)^2\right]^{-1} = x + iy$, then what is the value of $x + y$?

10. If the magnitude of difference of two unit vectors is $\sqrt{3}$, then the magnitude of sum of the two vectors is

11. Let $A(2, -4, 5)$ and $B(-2, 1, 2)$. What are the direction cosines of the line parallel to the line joining points $A$ and $B$?

12. If $\sin(A+B) = 1$ and $\cos(A-B) = \frac{\sqrt{3}}{2}$, then find the angle of $A$ and $B$:

13. If $\sin x + \sin y + \sin z = 3$, then $\cos 4x + \cos 4y + \cos 4z$ is

14. The value of $\sin^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right)$ is equal to

15. If the angles of a triangle $ABC$ are in the ratio $1 : 2 : 3$, then the corresponding sides are in the ratio

16. A straight line cuts off an intercept of 2 units on the positive direction of X-axis and passes through the point $(-3, 5)$. What is the foot of the perpendicular drawn from the point $(3, 3)$ on this line?

17. The equation $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$ represents a circle whose centre is

18. If $t_1$ and $t_2$ be the parameters of the end points of a focal chord for the parabola $y^2 = 4ax$, then which one is true?

19. The latus rectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is

20. If $t$ is a parameter, then $x = a\left(t + \frac{1}{t}\right), y = b\left(t - \frac{1}{t}\right)$ represents

21. The $\lim_{x \to 2} \frac{5}{\sqrt{2} - \sqrt{x}}$ is

22. For the function $f(x) = e^{\cos x}$, Rolle's theorem is

23. The function $f(x) = e^{ax} + e^{-ax}$, $a > 0$ is monotonically increasing for

24. Let $f(x) = \tan^{-1}x$. Then $f'(x) + f''(x) = 0$, when $x$ is equal to

25. If the rate of increase of the radius of a circle is $5\text{ cm/sec}$, then the rate of increase of its area when the radius is $20\text{ cm}$ will be

26. The value of the integral $\int \frac{dx}{(e^x + e^{-x})^2}$ is

27. What is the area bounded between the parabola $y^2 = x$ and the vertical line $x = 1$, in the first quadrant?

28. $\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} \, dx =$

29. For two events $A$ and $B$, it is given that $P(A) = 3/5$, $P(B) = 3/10$ and $P(A/B) = 2/3$. If $\bar{A}$ and $\bar{B}$ are the complementary events of $A$ and $B$, then what is $P(\bar{A}/\bar{B})$ equal to?

30. If the covariance between $x$ and $y$ is 30, variance of $x$ is 25 and variance of $y$ is 144, then what is the correlation coefficient?

Test Result