Class/Course  Engineering Entrance
Subject  Mathematics
Total Number of Question/s  3005
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1. Sets, Relations and Functions  Quiz
1. Let W denotes the words in the English dictionary. Define the relation R by
R = {(x,y) ∈ W x W : the words x and y have at least one letter in common).Then , R is
a) reflexive, symmetric and not transition
b) reflexive, symmetric and transitive
c) reflexive, not symmetric and transitive
d) not reflexive, symmetric and transitive
2. The function f(x) = $log\left (x + \sqrt{x^{2} + 1} \right )$, is
a) an even function
b) an odd function
c) a periodic function
d) neither an even nor an odd function

2. Complex Numbers and Quadratic Equations  Quiz
1. Let α, β be real and z be a complex number. If z^{2} + az + β = 0 has two distinct roots on the line Re z = 1, then it is necessary that
a) β ∈ (1,0)
b) β = 1
c) β ∈ (1,∞)
d) β ∈ (0,1)
2. The number of the real solutions of the equation x^{2}  3x + 2 = 0
a) 2
b) 4
c) 1
d) 3

3. Matrices and Determinants  Quiz
1. 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}
2. If (ω ≠ 1) is a cube root of unity, then $\begin{vmatrix} 1 &1 + i + \omega^{2} & 1\\ 1i& 1 & \omega^{2}1\\ i & 1+\omega  i &1 \end{vmatrix}$ equals
a) 0
b) 1
c) i
d) ω

4. Permutations and Combinations  Quiz
1. Statement I The number of ways distributing 10 identical balls in 4 distinct boxes such that no box is empty is ^{9}C_{3}.
statement II The number of ways of choosing any 3 places from 9 different places is ^{9}C_{3}.
a) Statem,ent I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
b) Statement I is true,Statement I is false.
c) Statement I is false,Statement I is true.
d) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
2. At an election , a voter may vote for any number of candidates not greater than the number to be elected. There are 10 candidates and 4 are to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote, is
a) 6210
b) 385
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 each natural number n,(n+1)^{7}  n^{7}  1 is divisible by 7.
Statement II For each natural number n, n^{7}  n is divisible by 7.
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. The remainder left out when 8^{2n}  (62)^{2n+1} is divided by 9 is
a) 0
b) 2
c) 7
d) 8
2. 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.

7. Sequences and Series  Quiz
1. If x = $\sum _{n=0}^{\infty }a^{n}, y = \sum_{n=0}^{\infty}b^{n}, z = \sum_{n =0}^{\infty}c^{n}$ where a ,b , c are in AP and a < 1, b < 1, c < 1, then x, y, z are in
a) HP
b) Arithematico  Geometric Progression
c) AP
d) GP
2. The sum of the first n terms of the series $1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + 2.6^{2} + ........$ is $\frac{n\left (n+1 \right )^{2}}{2}$ when n is even. When n is odd the sum is
a) $\frac{3n\left (n+1 \right )}{2}$
b) $\frac{n^{2}\left (n+1 \right )}{2}$
c) $\frac{n\left (n+1 \right )^{2}}{4}$
d) $\left [\frac{n\left (n+1 \right )}{2} \right ]^{2}$

8. Limits, Continuity and Differentiabilty  Quiz
1. If $lim_{x \rightarrow 0 }\frac{log\left (3 + x \right )  log\left (3  x \right )}{x}$ = k, the value of k is
a) 0
b) 1/3
c) 2/3
d) 2/3
2. A function is matched below against an interval where it is supposed to be increasing . Which of the following pairs is incorrectly matched?
a) Interval  (∞, 4), Function = x^{3} + 6x^{2} + 6
b) Interval = $\left (infty, \frac{1}{3} \right ]$, Function = 3x^{2}  2x + 1
c) Interval = [2,∞), Function = 2x^{3}  3x^{2}  12x + 6
d) Interval = (∞, ∞), Function = x^{3}  3x^{2} + 3x + 3

9. Integral Calculas  Quiz
1. Evaluate $\int_{0}^{\pi/2} \frac{\sqrt{sin x}}{\sqrt{sin x} + \sqrt{cos x}}dx$
a) $\frac{\pi}{4}$
b) $\frac{\pi}{2}$
c) 0
d) 1
2. $\int \left \{ \frac{\left (log x  1 \right )}{1 + \left (log x \right )^{2}} \right \}^{2}dx$ is equal to
a) $\frac{x}{\left (log x \right )^{2} + 1} + c$
b) $\frac{xe^{x}}{1 + x^{2}} + c$
c) $\frac{x}{x^{2} + 1} + c$
d) $\frac{log x}{\left (log x \right )^{2} + 1} + c$

10. Differential Equations  Quiz
1. Let I be the purchase value of an equipment and v(t) be the value after it has been used for t years . The value V(t) depreciates at a rate given by differential equation $\frac{dV\left (t \right )}{dt}$ = k(T t), where k > 0 is a constant and T is the total life in years of the equipment . Then , the scrap value V(T) of the equipment is
a) $I  \frac{kT^{2}}{2}$
b) $I  \frac{k\left (T  t \right )^{2}}{2}$
c) e^{kT}
d) $T^{2}  \frac{1}{k}$
2. The degree and order of the differential equation of the family of all parabolas whose axis is xaxis, are respectively
a) 2,1
b) 1,2
c) 3,2
d) 2,3

11. Coordinate Geometry  Quiz
1. The equation of a tangent to the parabola y^{2} = 8x is y = x + 2. The point on this from which the other tangent to the parabola is perpendicular to the given tangent is
a) (1,1)
b) (0,2)
c) (2,4)
d) (2,0)
2. The equation of the circle passing through the point (1,0) and (0,1) and having the smallest radius is
a) x^{2} + y^{2} + x + y  2 = 0
b) x^{2} + y^{2}  2x  2y + 1 = 0
c) x^{2} + y^{2}  x  y = 0
d) x^{2} + y^{2} + 2x + 2y  7 = 0

12. Three Dimensional Geometry  Quiz
1. The line passing through the points (5,1,a) and (3,b,1) crosses the yzplane at the point $\left (0, \frac{17}{2}, \frac{13}{2} \right )$. Then,
a) a = 8, b = 2
b) a = 2, b = 8
c) a = 4, b = 6
d) a = 6, b = 4
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 $\vec{a}, \vec{b}$ and $\vec{c}$ be nonzero vectors such that $\left (\vec{a} \times \vec{b} \right ) \times \vec{c}$ = $\frac{1}{3}\left  \vec{b} \right  \left  \vec{c} \right  \vec{a}$ . If θ is the acute angle between the vectors $\vec{b}$ and $\vec{c}$ , then sinθ equals
a) $\frac{1}{3}$
b) $\frac{\sqrt{2}}{3}$
c) $\frac{2}{3}$
d) $\frac{2\sqrt{2}}{3}$
2. If the vectors $p\hat{i} + \hat{j} + \hat{k}, \hat{i} + q\hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + r\hat{k}\left (p \ne q \ne r \ne 1) \right )$ are coplanar, then the value of pqr  ( p + q + r) is
a) 2
b) 2
c) 0
d) 1

14. Statistics and Probabilty  Quiz
1. For two two data sets, each of size, 5 the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is
a) $\frac{5}{2}$
b) $\frac{11}{2}$
c) 6
d) $\frac{13}{2}$
2. The average marks of boys in a class is 52 and that pof girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is
a) 40%
b) 20%
c) 80%
d) 60%

15. Trigonometry  Quiz
1. If sin(α + β) = 1, sin (α  β) = $\frac{1}{2}$ , then tan(α + 2β)tan(2α + β) is equal to
a) 1
b) 1
c) zero
d) None of these
2. If $cos^{1}x  cos^{1}\frac{y}{2}$ = $\alpha$ then 4x^{2}  4xy cos α + y^{2} is equal to
a) 4sin^{2}α
b) 4 sin^{2}α
c) 4
d) 2 sin 2α

16. Mathematical Reasoning  Quiz
1. Consider the following statements
P : Suman is brilliant.
Q : Suman is rich.
R : Suman is honest.
The negative of the statement. Suman is brilliant and dishonest if and only if Suman is rich can be expressed as
a) ∼ (Q ↔ (O P ∼ R)
b) ∼ Q ↔ P ^ R
c) ∼ (P ^ ∼ R) ↔ Q
d) ∼ P ^ (Q ↔ ∼ R)
2. The statement p → (q → p) is equivalent to
a) p → (p ↔ q)
b) p → (p → q)
c) p → (p ∨ q)
d) p → (p ∧ q)