Math Practice MCQs

Mathematics - Set Theory

1. Let a set $A$ have $n$ elements. If the total number of its subsets is 64, then $n$ is

A.5B.6C.7D.8

Mathematics - Relations and Functions

2. The relation $R$ defined on the set of natural numbers $\mathbb{N}$ as $xRy$ if $x$ divides $y$ is

A.Reflexive and symmetricB.Reflexive and transitiveC.Symmetric and transitiveD.Equivalence relation

Mathematics - Quadratic Equations

3. If the roots of the quadratic equation $x^2 - px + q = 0$ differ by 1, then the relation between $p$ and $q$ is

A.$p^2 = 4q + 1$B.$p^2 = 4q - 1$C.$q^2 = 4p + 1$D.$q^2 = 4p - 1$

Mathematics - Sequences and Series

4. The sum of the first $n$ terms of an arithmetic progression is $3n^2 + 5n$. Which term of this AP is 164?

A.$26^{\text{th}}$B.$27^{\text{th}}$C.$28^{\text{th}}$D.$29^{\text{th}}$

Mathematics - Binomial Theorem

5. The constant term in the binomial expansion of $(x - 1/x)^{10}$ is

A.252B.-252C.126D.-126

Mathematics - Matrices and Determinants

6. If $A = \begin{pmatrix} \alpha & 0 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 5 & 1 \end{pmatrix}$, and $A^2 = B$, then the value of $\alpha$ is

A.1B.-1C.4D.No real value exists

Mathematics - Matrices and Determinants

7. The system of equations $x + y + z = 1$, $2x + 3y + 2z = 2$, $ax + ay + 2az = 4$ will have a unique solution if

A.$a = 0$B.$a \ne 0$C.$a = 1$D.$a \ne 1$

Mathematics - Permutations and Combinations

8. In how many ways can a committee of 5 members be formed out of 7 men and 3 women such that exactly 2 women are included?

A.105B.140C.210D.35

Mathematics - Complex Numbers

9. If $z = \frac{1 + i}{1 - i}$, then $z^4$ equals

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

Mathematics - Vectors

10. The angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\sqrt{3}$ and $4$ respectively, and $\vec{a} \cdot \vec{b} = 2\sqrt{3}$ is

A.$\pi/6$B.$\pi/3$C.$\pi/4$D.$\pi/2$

Mathematics - 3D Geometry

11. The angle between the two lines whose direction ratios are $1, 1, 2$ and $\sqrt{3}-1, -\sqrt{3}-1, 4$ is

A.$30^\circ$B.$45^\circ$C.$60^\circ$D.$90^\circ$

Mathematics - Trigonometry

12. The value of $\tan 75^\circ - \cot 75^\circ$ is

A.$2\sqrt{3}$B.$2 + \sqrt{3}$C.$2 - \sqrt{3}$D.$4\sqrt{3}$

Mathematics - Trigonometry

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

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

Mathematics - Inverse Trigonometry

14. The value of $\cos\left(\sin^{-1}\frac{3}{5} + \cos^{-1}\frac{4}{5}\right)$ is

A.0B.1C.7/25D.-7/25

Mathematics - Trigonometry

15. In a $\triangle ABC$, if $a = 2$, $b = 3$ and $\sin A = 2/3$, then $\angle B$ is equal to

A.$30^\circ$B.$60^\circ$C.$90^\circ$D.$120^\circ$

Mathematics - Coordinate Geometry

16. The distance between the parallel lines $3x - 4y + 7 = 0$ and $3x - 4y + 12 = 0$ is

A.1B.2C.5D.1/5

Mathematics - Circles

17. The radius of the circle passing through the points $(0, 0)$, $(a, 0)$ and $(0, b)$ is

A.$\sqrt{a^2 + b^2}$B.$\frac{1}{2}\sqrt{a^2 + b^2}$C.$a^2 + b^2$D.$\frac{1}{2}(a + b)$

Mathematics - Parabola

18. The vertex of the parabola $y^2 - 4y - x + 3 = 0$ is

A.$(-1, 2)$B.$(1, -2)$C.$(-1, -2)$D.$(1, 2)$

Mathematics - Ellipse

19. If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is

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

Mathematics - Conic Sections

20. The asymptotes of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ are given by the equation

A.$2x \pm 3y = 0$B.$3x \pm 2y = 0$C.$2x + y = 0$D.$x \pm 2y = 0$

Mathematics - Calculus

21. The value of $\lim_{x \to 0} \frac{1 - \cos 2x}{x^2}$ is

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

Mathematics - Calculus

22. For the function $f(x) = x^2 - 4x + 3$ on the interval $[1, 3]$, the value of $c$ satisfying Rolle's theorem is

A.1.5B.2.0C.2.5D.4/3

Mathematics - Calculus

23. The function $f(x) = x^3 - 3x^2 + 3x - 100$ is

A.Monotonically increasing everywhereB.Monotonically decreasing everywhereC.Increasing for $x > 1$ and decreasing for $x < 1$D.Decreasing for $x > 1$ and increasing for $x < 1$

Mathematics - Calculus

24. If $y = A\sin x + B\cos x$, then $\frac{d^2y}{dx^2} + y$ is equal to

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

Mathematics - Application of Derivatives

25. The rate of change of the volume of a sphere with respect to its surface area when the radius is $2\text{ cm}$ is

A.$1\text{ cm}$B.$2\text{ cm}$C.$3\text{ cm}$D.$4\text{ cm}$

Mathematics - Integral Calculus

26. The value of the integral $\int e^x (f(x) + f'(x)) \, dx$ is equal to

A.$e^x f(x) + C$B.$e^x f'(x) + C$C.$e^{-x} f(x) + C$D.$f(x) + C$

Mathematics - Integral Calculus

27. The area of the region bounded by the curve $y = \sin x$, the x-axis, and the lines $x = 0$ and $x = \pi$ is

A.$1\text{ sq unit}$B.$2\text{ sq units}$C.$\pi\text{ sq units}$D.$\pi/2\text{ sq units}$

Mathematics - Definite Integrals

28. The value of $\int_{-\pi/2}^{\pi/2} \sin^3 x \cos^2 x \, dx$ is

A.1B.0C.$\pi/4$D.$\pi/2$

Mathematics - Probability

29. If $A$ and $B$ are independent events such that $P(A) = 0.3$ and $P(B) = 0.4$, then $P(A \cup B)$ is

A.0.12B.0.70C.0.58D.0.42

Mathematics - Statistics

30. If the variance of a data set is 16, what is its standard deviation?

A.256B.8C.4D.16

1. Let a set $A$ have $n$ elements. If the total number of its subsets is 64, then $n$ is

2. The relation $R$ defined on the set of natural numbers $\mathbb{N}$ as $xRy$ if $x$ divides $y$ is

3. If the roots of the quadratic equation $x^2 - px + q = 0$ differ by 1, then the relation between $p$ and $q$ is

4. The sum of the first $n$ terms of an arithmetic progression is $3n^2 + 5n$. Which term of this AP is 164?

5. The constant term in the binomial expansion of $(x - 1/x)^{10}$ is

6. If $A = \begin{pmatrix} \alpha & 0 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 5 & 1 \end{pmatrix}$, and $A^2 = B$, then the value of $\alpha$ is

7. The system of equations $x + y + z = 1$, $2x + 3y + 2z = 2$, $ax + ay + 2az = 4$ will have a unique solution if

8. In how many ways can a committee of 5 members be formed out of 7 men and 3 women such that exactly 2 women are included?

9. If $z = \frac{1 + i}{1 - i}$, then $z^4$ equals

10. The angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\sqrt{3}$ and $4$ respectively, and $\vec{a} \cdot \vec{b} = 2\sqrt{3}$ is

11. The angle between the two lines whose direction ratios are $1, 1, 2$ and $\sqrt{3}-1, -\sqrt{3}-1, 4$ is

12. The value of $\tan 75^\circ - \cot 75^\circ$ is

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

14. The value of $\cos\left(\sin^{-1}\frac{3}{5} + \cos^{-1}\frac{4}{5}\right)$ is

15. In a $\triangle ABC$, if $a = 2$, $b = 3$ and $\sin A = 2/3$, then $\angle B$ is equal to

16. The distance between the parallel lines $3x - 4y + 7 = 0$ and $3x - 4y + 12 = 0$ is

17. The radius of the circle passing through the points $(0, 0)$, $(a, 0)$ and $(0, b)$ is

18. The vertex of the parabola $y^2 - 4y - x + 3 = 0$ is

19. If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is

20. The asymptotes of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ are given by the equation

21. The value of $\lim_{x \to 0} \frac{1 - \cos 2x}{x^2}$ is

22. For the function $f(x) = x^2 - 4x + 3$ on the interval $[1, 3]$, the value of $c$ satisfying Rolle's theorem is

23. The function $f(x) = x^3 - 3x^2 + 3x - 100$ is

24. If $y = A\sin x + B\cos x$, then $\frac{d^2y}{dx^2} + y$ is equal to

25. The rate of change of the volume of a sphere with respect to its surface area when the radius is $2\text{ cm}$ is

26. The value of the integral $\int e^x (f(x) + f'(x)) \, dx$ is equal to

27. The area of the region bounded by the curve $y = \sin x$, the x-axis, and the lines $x = 0$ and $x = \pi$ is

28. The value of $\int_{-\pi/2}^{\pi/2} \sin^3 x \cos^2 x \, dx$ is

29. If $A$ and $B$ are independent events such that $P(A) = 0.3$ and $P(B) = 0.4$, then $P(A \cup B)$ is

30. If the variance of a data set is 16, what is its standard deviation?

Test Result