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) = $\frac{y3}{4}$
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 x^{2}  3x + 2 = 0
a) 2
b) 4
c) 1
d) 3
2. If (1p) is a root of quadratic equation x^{2} + px + (1p) = 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, ω^{2} 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) ω^{2}

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) 3^{8}
d) ^{8}C_{3}
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) $A^{n}$ = $2^{n1}A + \left (n1 \right )I$
b) $A^{n}$ = $nA + \left (n1 \right )I$
c) A^{n} = $2^{n1}A\left (n1 \right )I$
d) A^{n} = nA  (n1)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 $\sum_{r = 0}^{n}\left (r + 1 \right ),^{n}C_{r}$ = $\left (n+2 \right )2^{n1}$
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 S_{n} = $\sum_{r = 0}^{n}\frac{1}{^{n}C_{r}} \ and \ t_{n}= \sum_{r=0}^{n}\frac{r}{^{n}C_{r}} \ then \ \frac{t_{n}}{S_{n}}$ is equal to
a) $\frac{n}{2}$
b) $\frac{n}{2}$  1
c) n1
d) $\frac{2n1}{2}$

7. Sequences and Series  Quiz
1. If 1, $log_{3}\sqrt{\left (3^{1x} + 2 \right )}, log_{3}\left (4.3^{x}  1 \right )$ are in, Ap. Then x equals
a) log_{3}4
b) 1  log_{3}4
c) 1  log_{4}3
d) log_{4}3
2. If a_{1}, a_{2} , .....,a_{n} are in HP. Then the expression a_{1}a_{2} + a_{2}a_{3} + .... + a_{n1}a_{n} is equal to
a) (n1)(a_{1}  a_{n})
b) na_{1}a_{n}
c) (n1)a_{1}  a_{2}
d) n(a_{1}  a_{n})

8. Limits, Continuity and Differentiabilty  Quiz
1. Let f(x) = xx 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 y^{2} = 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 xaxis. 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 (x_{1}, y_{1}) and (x_{2},y_{2}) on the rectangular hyperbola xy = c^{2} 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 $\frac{x2}{1}$=$\frac{y3}{1}$=$\frac{z4}{k}$ and $\frac{x1}{k}$=$\frac{y4}{2}$=$\frac{z4}{1}$ 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 $\frac{x}{1}=\frac{y1}{2}=\frac{z2}{3}$
Statement II The line the line segment joining $\frac{x}{1}=\frac{y1}{2}=\frac{z2}{3}$ 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 noncoplanar 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 noncoplanar 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.