Math Practice MCQs

Mathematics - Set Theory

1. Three sets $A$, $B$, $C$ are such that $A = B \cap C$ and $B = C \cap A$, then

A.$A \subset B$B.$A \supset B$C.$A \equiv B$D.$A \subset B'$

Mathematics - Logarithms

2. The value of $\left(\frac{1}{\log_3 12} + \frac{1}{\log_4 12}\right)$ is

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

Mathematics - Sequences and Series

3. If three positive real numbers $a, b, c$ are in A.P. and $abc = 4$, then the minimum possible value of $b$ is

A.$2^{3/2}$B.$2^{2/3}$C.$2^{1/3}$D.$2^{5/2}$

Mathematics - Quadratic Equations

4. If $\alpha$ and $\beta$ are roots of the equation $x^2 + 2x + 3 = 0$ and $S_n = \alpha^n + \beta^n$, then the value of $\frac{S_{10} + 3S_8}{S_9}$ is

A.-3B.-2C.-4D.-6

Mathematics - Permutations and Combinations

5. If $\binom{n-1}{3} + \binom{n-1}{4} > \binom{n}{3}$, then $n$ is greater than the integer

A.5B.6C.4D.7

Mathematics - Permutations and Combinations

6. How many odd numbers of six significant digits can be formed with the digits $1, 1, 2, 5, 6, 7$ when no digit is repeated beyond its given multiplicity?

A.120B.96C.360D.288

Mathematics - Binomial Theorem

7. If the coefficients of the $r^{\text{th}}$, $(r+1)^{\text{th}}$ and $(r+2)^{\text{th}}$ terms in the expansion of $(1 + x)^{14}$ are in A.P., then the value of $r$ is:

A.11B.5C.7D.6

Mathematics - Number Theory

8. For positive integer $n$, $n^3 + 2n$ is always divisible by

A.3B.7C.5D.6

Mathematics - Matrices and Determinants

9. If the matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is commutative with the matrix $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then

A.$a = 0, b = c$B.$b = 0, c = d$C.$c = 0, d = a$D.$d = 0, a = b$

Mathematics - Vectors

10. $ABCD$ is a parallelogram and $P$ is the point of intersection of the diagonals. If $O$ is the origin, then $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD}$ is equal to

A.$4\vec{OP}$B.$2\vec{OP}$C.$\vec{OP}$D.Null vector

Mathematics - 3D Geometry

11. The angle between a normal to the plane $2x - y + 2z - 1 = 0$ and the X-axis is

A.$\cos^{-1}\frac{2}{3}$B.$\cos^{-1}\frac{1}{5}$C.$\cos^{-1}\frac{3}{4}$D.$\cos^{-1}\frac{1}{3}$

Mathematics - Statistics

12. The arithmetic mean of $1, 8, 27, 64, \dots$ up to $n$ terms is given by

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

Mathematics - Probability

13. The probability of drawing a card that is not a heart from a standard pack of 52 cards is

A.1/2B.3/2C.3/4D.4/3

Mathematics - Trigonometry

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

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

Mathematics - Inverse Trigonometry

15. If $\cos^{-1} x - \cos^{-1}\frac{y}{2} = \alpha$, then $4x^2 - 4xy\cos\alpha + y^2$ is

A.$-4\sin^2\alpha$B.$4\sin^2\alpha$C.4D.$2\sin 2\alpha$

Mathematics - Trigonometry

16. From the top of a lighthouse, $100\text{ m}$ high, the angle of depression of a boat is $\tan^{-1}\left(\frac{5}{12}\right)$. What is the distance between the boat and the lighthouse?

A.$120\text{ m}$B.$180\text{ m}$C.$240\text{ m}$D.$360\text{ m}$

Mathematics - Trigonometry

17. In a triangle $ABC$, with usual notations, $(a + b + c)(a + b - c) = 3ab$, then $\angle C =$

A.$\frac{\pi}{2}$B.$\frac{\pi}{4}$C.$\frac{\pi}{3}$D.$\frac{\pi}{6}$

Mathematics - Complex Numbers

18. If $|z - 2i| \le \sqrt{2}$, then the maximum value of $|3 + i(z - 1)|$ is

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

Mathematics - Complex Numbers

19. The value of $(1 - \omega + \omega^2)^5 + (1 - \omega - \omega^2)^5$, where $\omega$ and $\omega^2$ are the complex cube roots of unity is

A.0B.$32\omega$C.-32D.32

Mathematics - Functions

20. The domain of definition of the function $f(x)$ given by the equation $2^x + 2^y = 2$ is

A.$0 < x \le 1$B.$0 \le x \le 1$C.$-\infty < x \le 0$D.$-\infty < x < 1$

Mathematics - Functions

21. If $f(x) = x^2 + 1$ and $g(x) = \frac{1}{x}$, then the value of $f(g(g(f(x))))$ at $x = 1$ is

A.4B.1C.5D.3

Mathematics - Calculus

22. The approximate value of $x^3 - 2x^2 + 3x + 2$ at $x = 2.01$ is

A.8.07B.8.27C.8.007D.8.17

Mathematics - Calculus

23. Derivative of $y = (\sin x)^{(\sin x)^{(\sin x)^{\dots \infty}}}$ is

A.$\frac{y^2 \cot x}{1 + y\log(\sin x)}$B.$\frac{y^2 \tan x}{1 + y\log(\sin x)}$C.$\frac{y^2 \cot x}{1 - y\log(\sin x)}$D.$\frac{y^2 \tan x}{1 - y\log(\sin x)}$

Mathematics - Integral Calculus

24. $\int 2x \sin 2x \, dx =$

A.$\frac{\sin 2x}{2} + x\cos 2x + c$B.$\frac{\sin 2x}{2} + \cos 2x + c$C.$\frac{\cos 2x}{2} - x\cos 2x + c$D.$\frac{\sin 2x}{2} - x\cos 2x + c$

Mathematics - Definite Integrals

25. $\int_{3/4}^{5/4} \sqrt{\left(x - \frac{3}{4}\right)\left(\frac{5}{4} - x\right)} \, dx =$

A.$\frac{\pi}{24}$B.$\pi$C.$\frac{\pi}{12}$D.$\frac{\pi}{32}$

Mathematics - Integral Calculus

26. Find the area enclosed by curve $y = \sqrt{3}x + 3$ and $(y - 3)^2 = 4x$

A.$\frac{8}{3\sqrt{3}}$B.2/3C.$\frac{8}{9\sqrt{3}}$D.6

Mathematics - Differential Equations

27. The general solution of the differential equation $\frac{dy}{dx} = e^{y+x} + e^{y-x}$ where $c$ is an arbitrary constant is

A.$e^{-y} = e^x - e^{-x} + c$B.$-e^{-y} = e^x - e^{-x} + c$C.$-e^{-y} = e^x + e^{-x} + c$D.$e^y = e^x + e^{-x} + c$

Mathematics - Coordinate Geometry

28. Given the straight line equation: $x + y = 1$. If the origin is shifted to $(1, 2)$, what is the new equation of the line in the new coordinate system?

A.$x' + y' = 1$B.$x' + y' = -2$C.$x' + y' = 4$D.$x' + y' = 2$

Mathematics - Conic Sections

29. The equation of the ellipse having vertices at $(\pm 5, 0)$ and foci $(\pm 4, 0)$ is

A.$\frac{x^2}{25} + \frac{y^2}{16} = 1$B.$9x^2 + 25y^2 = 225$C.$\frac{x^2}{9} + \frac{y^2}{25} = 1$D.$4x^2 + 5y^2 = 20$

Mathematics - Real Numbers

30. How many real numbers lie in the interval $0 < x \le 1$?

A.InfiniteB.FiniteC.0D.1

1. Three sets $A$, $B$, $C$ are such that $A = B \cap C$ and $B = C \cap A$, then

2. The value of $\left(\frac{1}{\log_3 12} + \frac{1}{\log_4 12}\right)$ is

3. If three positive real numbers $a, b, c$ are in A.P. and $abc = 4$, then the minimum possible value of $b$ is

4. If $\alpha$ and $\beta$ are roots of the equation $x^2 + 2x + 3 = 0$ and $S_n = \alpha^n + \beta^n$, then the value of $\frac{S_{10} + 3S_8}{S_9}$ is

5. If $\binom{n-1}{3} + \binom{n-1}{4} > \binom{n}{3}$, then $n$ is greater than the integer

6. How many odd numbers of six significant digits can be formed with the digits $1, 1, 2, 5, 6, 7$ when no digit is repeated beyond its given multiplicity?

7. If the coefficients of the $r^{\text{th}}$, $(r+1)^{\text{th}}$ and $(r+2)^{\text{th}}$ terms in the expansion of $(1 + x)^{14}$ are in A.P., then the value of $r$ is:

8. For positive integer $n$, $n^3 + 2n$ is always divisible by

9. If the matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is commutative with the matrix $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then

10. $ABCD$ is a parallelogram and $P$ is the point of intersection of the diagonals. If $O$ is the origin, then $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD}$ is equal to

11. The angle between a normal to the plane $2x - y + 2z - 1 = 0$ and the X-axis is

12. The arithmetic mean of $1, 8, 27, 64, \dots$ up to $n$ terms is given by

13. The probability of drawing a card that is not a heart from a standard pack of 52 cards is

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

15. If $\cos^{-1} x - \cos^{-1}\frac{y}{2} = \alpha$, then $4x^2 - 4xy\cos\alpha + y^2$ is

16. From the top of a lighthouse, $100\text{ m}$ high, the angle of depression of a boat is $\tan^{-1}\left(\frac{5}{12}\right)$. What is the distance between the boat and the lighthouse?

17. In a triangle $ABC$, with usual notations, $(a + b + c)(a + b - c) = 3ab$, then $\angle C =$

18. If $|z - 2i| \le \sqrt{2}$, then the maximum value of $|3 + i(z - 1)|$ is

19. The value of $(1 - \omega + \omega^2)^5 + (1 - \omega - \omega^2)^5$, where $\omega$ and $\omega^2$ are the complex cube roots of unity is

20. The domain of definition of the function $f(x)$ given by the equation $2^x + 2^y = 2$ is

21. If $f(x) = x^2 + 1$ and $g(x) = \frac{1}{x}$, then the value of $f(g(g(f(x))))$ at $x = 1$ is

22. The approximate value of $x^3 - 2x^2 + 3x + 2$ at $x = 2.01$ is

23. Derivative of $y = (\sin x)^{(\sin x)^{(\sin x)^{\dots \infty}}}$ is

24. $\int 2x \sin 2x \, dx =$

25. $\int_{3/4}^{5/4} \sqrt{\left(x - \frac{3}{4}\right)\left(\frac{5}{4} - x\right)} \, dx =$

26. Find the area enclosed by curve $y = \sqrt{3}x + 3$ and $(y - 3)^2 = 4x$

27. The general solution of the differential equation $\frac{dy}{dx} = e^{y+x} + e^{y-x}$ where $c$ is an arbitrary constant is

28. Given the straight line equation: $x + y = 1$. If the origin is shifted to $(1, 2)$, what is the new equation of the line in the new coordinate system?

29. The equation of the ellipse having vertices at $(\pm 5, 0)$ and foci $(\pm 4, 0)$ is

30. How many real numbers lie in the interval $0 < x \le 1$?

Test Result