Math Practice MCQs

Mathematics - Quadratic Equations

1. If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x + 1 = 0$, then the value of $\alpha^2 + \beta^2$ is:

A.1.25B.1.5C.0.5D.2

Mathematics - Number Theory

2. The last digit of $7^{105}$ is:

A.1B.3C.7D.9

Mathematics - Conic Sections

3. The eccentricity of the hyperbola $16x^2 - 9y^2 = 144$ is:

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

Mathematics - Matrices and Determinants

4. If $A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, then the determinant of $A$ is:

A.0B.1C.-1D.$\cos 2\theta$

Mathematics - Sequences and Series

5. The sum of the infinite geometric series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ is:

A.1.5B.2C.2.5D.Infinity

Mathematics - Complex Numbers

6. If $z = x + iy$ and $|z - 1| = |z + 1|$, then the locus of $z$ is:

A.x-axisB.y-axisC.A circleD.A parabola

Mathematics - Vectors

7. For any two non-zero vectors $\vec{a}$ and $\vec{b}$, the value of $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2$ is equal to:

A.$|\vec{a}|^2 + |\vec{b}|^2$B.$|\vec{a}|^2|\vec{b}|^2$C.0D.1

Mathematics - Trigonometry

8. The maximum value of $f(x) = \sin x \cos x$ is:

A.1B.0.5C.2D.$\sqrt{2}$

Mathematics - Calculus

9. If $x^2 + y^2 = a^2$, then $\frac{dy}{dx}$ is equal to:

A.$\frac{y}{x}$B.$-\frac{y}{x}$C.$\frac{x}{y}$D.$-\frac{x}{y}$

Mathematics - Inverse Trigonometry

10. The principal value of $\sin^{-1}\left(\sin\frac{2\pi}{3}\right)$ is:

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

Mathematics - Calculus

11. The derivative of $y = e^{\sin x}$ with respect to $x$ is:

A.$e^{\sin x}$B.$e^{\cos x}$C.$e^{\sin x}\cos x$D.$e^{\sin x}\sin x$

Mathematics - Calculus

12. What is the derivative of $\ln(x^2 + 1)$ with respect to $x$?

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

Mathematics - Binomial Theorem

13. In the binomial expansion of $\left(x + \frac{1}{x}\right)^{10}$, the term independent of $x$ is:

A.120B.210C.252D.10

Mathematics - Probability

14. Two dice are thrown simultaneously. The probability of getting a sum greater than 9 is:

A.1/6B.1/12C.5/36D.1/9

Mathematics - Calculus

15. Evaluate $\lim_{x \to 0} \frac{1 - \cos 2x}{x^2}$:

A.0B.1C.2D.4

Mathematics - Calculus

16. $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$ is equal to:

A.1B.0C.$e$D.Infinity

Mathematics - 3D Geometry

17. The equation of the plane passing through the point $(1, -2, 3)$ and parallel to the plane $2x - 3y + z = 5$ is:

A.$2x - 3y + z = 11$B.$2x - 3y + z = 0$C.$x - 2y + 3z = 5$D.$2x - 3y + z = 10$

Mathematics - Integral Calculus

18. The area of the region bounded by $y^2 = x$ and the line $x = 4$ is:

A.16/3 sq. unitsB.32/3 sq. unitsC.8/3 sq. unitsD.16 sq. units

Mathematics - Parabola

19. If the line $y = mx + 1$ is tangent to the parabola $y^2 = 4x$, then the value of $m$ is:

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

Mathematics - 3D Geometry

20. If a line makes angles $\alpha, \beta, \gamma$ with the positive direction of coordinate axes, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to:

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

Mathematics - Permutations and Combinations

21. How many different permutations can be made out of the letters of the word 'MATHEMATICS'?

A.$\frac{11!}{2!2!2!}$B.$11!$C.$\frac{11!}{2!}$D.$\frac{11!}{3!}$

Mathematics - Circles

22. The radius of the circle $x^2 + y^2 - 6x + 8y - 11 = 0$ is:

A.5B.6C.11D.36

Mathematics - Circles

23. The equation of the tangent to the circle $x^2 + y^2 = 25$ at the point $(3, 4)$ is:

A.$3x + 4y = 5$B.$4x + 3y = 25$C.$3x + 4y = 25$D.$3x - 4y = 25$

Mathematics - Logarithms

24. If $\log_2(\log_3 x) = 1$, then the value of $x$ is:

A.3B.6C.8D.9

Mathematics - Integral Calculus

25. $\int \frac{1}{x \ln x} \, dx$ is equal to:

A.$\ln x + C$B.$\ln(\ln x) + C$C.$x \ln x + C$D.$\frac{1}{\ln x} + C$

Mathematics - Calculus

26. The slope of the tangent to the curve $y = x^2 - 2x + 1$ at $x = 3$ is:

A.2B.4C.6D.8

Mathematics - 3D Geometry

27. The normal vector to the plane $3x - 4y + 5z = 10$ is:

A.$3\hat{i} + 4\hat{j} - 5\hat{k}$B.$3\hat{i} - 4\hat{j} + 5\hat{k}$C.$\hat{i} + \hat{j} + \hat{k}$D.$-3\hat{i} + 4\hat{j} + 5\hat{k}$

Mathematics - Differential Equations

28. The integrating factor of the linear differential equation $\frac{dy}{dx} + y\sec^2 x = \tan x \sec^2 x$ is:

A.$e^{\tan x}$B.$e^{\sec x}$C.$\sec^2 x$D.$\tan x$

Mathematics - Functions

29. The domain of the function $f(x) = \sin^{-1}(2x)$ is:

A.$[-1, 1]$B.$[-0.5, 0.5]$C.$[0, 1]$D.$[-\pi/2, \pi/2]$

Mathematics - Statistics

30. If the variance of a data set is 16, and every observation is multiplied by 3, the new standard deviation will be:

A.16B.4C.12D.144

1. If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x + 1 = 0$, then the value of $\alpha^2 + \beta^2$ is:

2. The last digit of $7^{105}$ is:

3. The eccentricity of the hyperbola $16x^2 - 9y^2 = 144$ is:

4. If $A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, then the determinant of $A$ is:

5. The sum of the infinite geometric series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ is:

6. If $z = x + iy$ and $|z - 1| = |z + 1|$, then the locus of $z$ is:

7. For any two non-zero vectors $\vec{a}$ and $\vec{b}$, the value of $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2$ is equal to:

8. The maximum value of $f(x) = \sin x \cos x$ is:

9. If $x^2 + y^2 = a^2$, then $\frac{dy}{dx}$ is equal to:

10. The principal value of $\sin^{-1}\left(\sin\frac{2\pi}{3}\right)$ is:

11. The derivative of $y = e^{\sin x}$ with respect to $x$ is:

12. What is the derivative of $\ln(x^2 + 1)$ with respect to $x$?

13. In the binomial expansion of $\left(x + \frac{1}{x}\right)^{10}$, the term independent of $x$ is:

14. Two dice are thrown simultaneously. The probability of getting a sum greater than 9 is:

15. Evaluate $\lim_{x \to 0} \frac{1 - \cos 2x}{x^2}$:

16. $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$ is equal to:

17. The equation of the plane passing through the point $(1, -2, 3)$ and parallel to the plane $2x - 3y + z = 5$ is:

18. The area of the region bounded by $y^2 = x$ and the line $x = 4$ is:

19. If the line $y = mx + 1$ is tangent to the parabola $y^2 = 4x$, then the value of $m$ is:

20. If a line makes angles $\alpha, \beta, \gamma$ with the positive direction of coordinate axes, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to:

21. How many different permutations can be made out of the letters of the word 'MATHEMATICS'?

22. The radius of the circle $x^2 + y^2 - 6x + 8y - 11 = 0$ is:

23. The equation of the tangent to the circle $x^2 + y^2 = 25$ at the point $(3, 4)$ is:

24. If $\log_2(\log_3 x) = 1$, then the value of $x$ is:

25. $\int \frac{1}{x \ln x} \, dx$ is equal to:

26. The slope of the tangent to the curve $y = x^2 - 2x + 1$ at $x = 3$ is:

27. The normal vector to the plane $3x - 4y + 5z = 10$ is:

28. The integrating factor of the linear differential equation $\frac{dy}{dx} + y\sec^2 x = \tan x \sec^2 x$ is:

29. The domain of the function $f(x) = \sin^{-1}(2x)$ is:

30. If the variance of a data set is 16, and every observation is multiplied by 3, the new standard deviation will be:

Test Result