## 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 z2 + 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 x2 - 3|x| + 2 = 0
a) 2
b) 4
c) 1
d) 3

• 3. Matrices and Determinants - Quiz

1. 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

2. If (ω ≠ 1) is a cube root of unity, then $\begin{vmatrix} 1 &1 + i + \omega^{2} & 1\\ 1-i& -1 & \omega^{2}-1\\ -i & -1+\omega - i &-1 \end{vmatrix}$ 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 9C3.
statement II The number of ways of choosing any 3 places from 9 different places is 9C3.
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) $A^{n}$ = $2^{n-1}A + \left (n-1 \right )I$
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 each natural number n,(n+1)7 - n7 - 1 is divisible by 7.
Statement II For each natural number n, n7 - 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 82n - (62)2n+1 is divided by 9 is
a) 0
b) 2
c) 7
d) 8

2. Statement I $\sum_{r = 0}^{n}\left (r + 1 \right ),^{n}C_{r}$ = $\left (n+2 \right )2^{n-1}$
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.

• 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 = x3 + 6x2 + 6
b) Interval = $\left (-infty, \frac{1}{3} \right ]$, Function = 3x2 - 2x + 1
c) Interval = [2,∞), Function = 2x3 - 3x2 - 12x + 6
d) Interval = (-∞, ∞), Function = x3 - 3x2 + 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 x-axis, 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 y2 = 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) x2 + y2 + x + y - 2 = 0
b) x2 + y2 - 2x - 2y + 1 = 0
c) x2 + y2 - x - y = 0
d) x2 + y2 + 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 yz-plane 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 $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$
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 $\vec{a}, \vec{b}$ and $\vec{c}$ be non-zero 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 4x2 - 4xy cos α + y2 is equal to
a) -4sin2α
b) 4 sin2α
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)