## Class/Course - Engineering Entrance

### Subject - Mathematics

#### Total Number of Question/s - 3005

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• 1. Sets, Relations and Functions - Quiz

1. Let f : N → Y be a function defined as f(x) = 4x + 3 where
Y = { y ∈ N : y = 4x + 3 for some x ∈ N}.
Show that f is invertible and its inverse is
a) g(y) = $\frac{y-3}{4}$
b) g(y) = $\frac{3y+4}{3}$
c) g(y) = $4 + \frac{y+3}{4}$
d) g(y) = $\frac{y+3}{4}$

2. Let R = {(1.3), (4,2), (4,2), (2,4), (2,3), (3,1)} be relation on the set A = {1,2,3,4} . The relation of R is
a) a function
b) transitive
c) not symmetric
d) reflexive

• 2. Complex Numbers and Quadratic Equations - Quiz

1. The number of the real solutions of the equation x2 - 3|x| + 2 = 0
a) 2
b) 4
c) 1
d) 3

2. If (1-p) is a root of quadratic equation x2 + px + (1-p) = 0, then its roots are
a) 0,1
b) -1,1
c) 0,-1
d) -1, 2

• 3. Matrices and Determinants - Quiz

1. Let A and b be two symmetric matrices of order 3.
Statement I A (BA) and (AB) A are symmetric matrices.
Statement II AB is symmetric matrix, if matrix multiplication of A and B is communitative.
a) Statement 1 is true, Statement II is true; Statement II is not a correct explanation for Statement I.
b) Statement I is true, Statement II is false.
c) Statement I is false, Statement II is true.
d) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.

2. If 1, ω, ω2 are the cube roots of unity, then Δ = $\begin{bmatrix} 1 & \omega^{n} & \omega^{2n} \\ \omega^{n} & \omega^{2n} &1 \\ \omega^{2n}& 1 & \omega^{n} \end{bmatrix}$ is equal to
a) 0
b) 1
c) ω
d) ω2

• 4. Permutations and Combinations - Quiz

1. The number of ways of distributing 8 identical balls in distinct boxes, so that none of the boxes is empty, is
a) 5
b) 21
c) 38
d) 8C3

2. If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary , then the word SACHIN appears at serial number
a) 602
b) 603
c) 600
d) 601

• 5. Mathematical Induction - Quiz

1. If A = $\begin{bmatrix} 1 &0 \\ 1& 1 \end{bmatrix}$ and I = $\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}$, then which one of the following holds for all n ≥ 1, by the principle of mathematical induction?
a) $A^{n}$ = $2^{n-1}A + \left (n-1 \right )I$
b) $A^{n}$ = $nA + \left (n-1 \right )I$
c) An = $2^{n-1}A-\left (n-1 \right )I$
d) An = nA - (n-1)I

2. Statement I For every natural number n ≥ 2
$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + .......... + \frac{1}{\sqrt{n}}$ > $\sqrt{n}$
statement II For every natural number $n \ge 2. \sqrt{n\left (n + 1 \right )} < + 1$
a) Statement I is false , Statement II is true.
b) Statement I is true, Statement II is true; Statement II is correct explanation for Statement I.
c) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
d) Statement I is true, Statement II is false.

• 6. Binomial Theorem - Quiz

1. Statement I $\sum_{r = 0}^{n}\left (r + 1 \right ),^{n}C_{r}$ = $\left (n+2 \right )2^{n-1}$
Statement II $\sum_{r=0}^{n}\left (r+1 \right )^{n}C_{r}.x^{r}$ = $\left (1+x \right )^{n} + nx\left (1+x \right )^{n+1}$
a) Statement I is false, Statement II is true.
b) Statement I is true, Statement II is true; Statement II is a correct explanations of Statement I.
c) Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I.
d) Statement I is true, Statement II is false.

2. If Sn = $\sum_{r = 0}^{n}\frac{1}{^{n}C_{r}} \ and \ t_{n}= \sum_{r=0}^{n}\frac{r}{^{n}C_{r}} \ then \ \frac{t_{n}}{S_{n}}$ is equal to
a) $\frac{n}{2}$
b) $\frac{n}{2}$ - 1
c) n-1
d) $\frac{2n-1}{2}$

• 7. Sequences and Series - Quiz

1. If 1, $log_{3}\sqrt{\left (3^{1-x} + 2 \right )}, log_{3}\left (4.3^{x} - 1 \right )$ are in, Ap. Then x equals
a) log34
b) 1 - log34
c) 1 - log43
d) log43

2. If a1, a2 , .....,an are in HP. Then the expression a1a2 + a2a3 + .... + an-1an is equal to
a) (n-1)(a1 - an)
b) na1an
c) (n-1)a1 - a2
d) n(a1 - an)

• 8. Limits, Continuity and Differentiabilty - Quiz

1. Let f(x) = x|x| and g(x) = sin x
Statement I gof is differentiable at x = 0 and its derivative is continuous at that point.
Statement II gof is twice differentiable at x = 0.
a) Statement I is false, Statement II is true.
b) Statement I is true, Statement II is is true, Statement II is a correct explanation for Statement I.
c) Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I.
d) Statement I is true, Statement II is false.

2. The function f : R/(0) → R given by
f(x) = $\frac{1}{x} - \frac{2}{e^{2x} - 1}$
can be made continuous at x = 0 by defining f(0) as
a) 2
b) -1
c) 0
d) 1

• 9. Integral Calculas - Quiz

1. $\int_{0}^{10x}\left | sin x\right|$ is
a) 20
b) 8
c) 10
d) 18

2. Let F(x) = f(x) + $f\left (\frac{1}{x} \right )$ , where f(x) = $\int_{1}^{x}\frac{log t}{1 + t}dt$. Then , F(e) equals
a) $\frac{1}{2}$
b) 0
c) 1
d) 2

• 10. Differential Equations - Quiz

1. The solution of the differential equation
$\left (1 + y^{2} \right ) + \left (x - e^{tan^{-1}y} \right )\frac{dy}{dx}$ = 0 is
a) $\left (x - 2 \right )$= $ce^{-2tan^{-1}y}$
b) $2xe^{tan^{-1}y}$ = $2^{tan^{-1}y} + c$
c) $xe^{tan^{-1}y}$ = $tan^{-1}y + c$
d) $xe^{2tan^{-1}y}$ = $e^{tan^{-1}y} + c$

2. The differential equation representating the family of curves y2 = 2c(x + $\sqrt{c}$ ), where c > 0, is a parameter, is of order and degree as follows
a) order 2, degree 2
b) order 1, degree 3
c) order 1, degree 1
d) order 1, degree 2

• 11. Coordinate Geometry - Quiz

1. A variable circle passes through the fixed point A(p,q) and touches x-axis. The locus of the other end of the diameter through A is
a) (x - p)2 = 4qy
b) (x - q)2 = 4py
c) (y - p)2 = 4qx
d) (y - q)2 = 4px

2. The equation of the chord joining two points (x1, y1) and (x2,y2) on the rectangular hyperbola xy = c2 is
a) $\frac{x}{x_{1} + x_{2}} + \frac{y}{y_{1} + y_{2}}$ = 1
b) $\frac{x}{x_{1} - x_{2}} + \frac{y}{y_{1} - y_{2}}$ = 1
c) $\frac{x}{y_{1} + y_{2}} + \frac{y}{x_{1} + x_{2}}$ = 1
d) $\frac{x}{y_{1} - y_{2}} + \frac{y}{x_{1} - x_{2}}$ = 1

• 12. Three Dimensional Geometry - Quiz

1. The line $\frac{x-2}{1}$=$\frac{y-3}{1}$=$\frac{z-4}{-k}$ and $\frac{x-1}{k}$=$\frac{y-4}{2}$=$\frac{z-4}{1}$ are coplanar, if
a) k = 0 or -1
b) k = 1 or -1
c) k = 0 or -3
d) k = 3 or -3

2. Statement I The point A(1,0,7) is the mirror image of the point B(1,6,3) in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$
Statement II The line the line segment joining $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ bisects the line segment joining A(1,0,7) and B(1,6,3).
a) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
b) Statement I is true, Statement II is false.
c) Statement I is false, Statement II is true.
d) Statement I is true, Statement II is true; Statement II is a correct Explanation for Statement I.

• 13. Vector Algebra - Quiz

1. Let $\vec{a}$ = $\hat{i} - \hat{k}$ , $\vec{b}$ = $x\hat{i} + \hat{j} + \left (1 - x \right )k^{2}$ and $\vec{c}$ = $y\hat{i} + x\hat{j} + \left (1 + x - y \right )\hat{k}$ . Then $\left [\vec{a}\vec{b}\vec{c} \right ]$ depends on
a) neither x notr y
b) both x and y
c) only x
d) only y

2. If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and λ be a real number, then the vectors $\vec{a} + 2\vec{b} + 3\vec{c}, \lambda \vec{b} + 4\vec{c}$ and $\left (2\lambda - 1 \right )\vec{c}$ are non-coplanar for
a) all value of λ
b) all except one value of λ
c) all except two values of λ
d) no value of λ

• 14. Statistics and Probabilty - Quiz

1. In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average of the girls?
a) 73
b) 65
c) 68
d) 74

2. A pair of fair dice is thrown in independently three times. The probability of getting a score of exactly 9 is twice is
a) 1/729
b) 8/9
c) 8/729
d) 8/243

• 15. Trigonometry - Quiz

1. The possible values of &thet; € (0,π) such that sin(θ) + sin(4θ) + sin(7θ) = 0 are
a) $\frac{2\pi }{9},\frac{\pi }{4},\frac{4\pi }{9},\frac{\pi}{2},\frac{3\pi}{4},\frac{8\pi}{9}$
b) $\frac{\pi }{4},\frac{5\pi }{12},\frac{\pi }{2},\frac{2\pi}{3},\frac{3\pi}{4},\frac{8\pi}{9}$
c) $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi}{2},\frac{3\pi}{4},\frac{35\pi}{36}$
d) $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi}{3},\frac{3\pi}{4},\frac{8\pi}{9}$

2. If f : R → S, defined by
f(x) = $sin x - \sqrt{3} cosx + 1$, is onto, then the interval of S is
a) [0,3]
b) [-1,1]
c) [0,1]
d) [-1,3]

• 16. Mathematical Reasoning - Quiz

1. The only statement among the following that is a tautology is
a) B → ∧ ( A → B)]
b) A ∧ (A ∨ B)
c) A ∨ (A ∧ B)
d) [A ∧ (A → B)] → B

2. Statement I $\sim$ (p ↔ ∼ q) is equivalent to p ↔ q.
Statement II ∼ (p ↔ ∼ q) is a tautology.
a) Statement I is true, Statement II is true; Statement II is true; Statement II is a correct explanation for Statement I.
b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
c) Statement I is true, Statement II is false.
d) Statement I is false, Statement II is true.