## Class/Course - Engineering Entrance

### Subject - Mathematics

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

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• 1. Sets, Relations and Functions - Quiz

1. Domain of definition of the function f(x) = $\frac{3}{4 - x^{2}} + log_{10}\left (x^{3} - x\right )$ , is
a) (1,2)
b) (-1,0) ∪ (1,2)
c) (1,2) ∪ (2,$\infty$)
d) (-1,0) ∪ (1,2) ∪ (2,$\infty$ )

2. Let for a ≠ a1 ≠ 0, f(x) = ax2 = ax2 + bx + c,
g(x) = a1x2 + b1 x + c1 and p(x) = f(x) - g(x) , If p(x) = 0 only for x = -1 and p(-2) = 2, then the value of p(2) is

a) 18
b) 3
c) 9
d) 6

• 2. Complex Numbers and Quadratic Equations - Quiz

1. Let z, w be complex numbers such that $\bar{z} + i\bar{w}$ = 0 and arg (zw) = π . Then arg (z) equals
a) $-\frac{\pi}{4}$
b) $-\frac{\pi}{2}$
c) $-\frac{3\pi}{4}$
d) $-\frac{5\pi}{4}$

2. If both the roots of the quadratic equation x2 - 2kx + k2 + k -5 = 0 are less than k lies in the interval
a) [4,5]
b) (-∞,4)
c) (6,∞)
d) (5,6]

• 3. Matrices and Determinants - Quiz

1. If the trivial solution is the only soluytion of the system of equations
x - ky + z = 0
k + 3y - kz = 0
3x + y - z = 0
Then the set of lal values of k is
a) {2,-3}
b) R-{2,-3}
c) R-{2}
d) R-{-3}

2. If l , m , n are the pth, qth and rth terms of an GP and all positive, then $\begin{bmatrix} 1 & \omega^{n} &\omega^{2n} \\ \omega^{n} & \omega^{2n} &1 \\ \omega^{2n} &1 & \omega^{n} \end{bmatrix}$ equals
a) 3
b) 2
c) 1
d) 0

• 4. Permutations and Combinations - Quiz

1. The number of ways in which 6 men and 15 women can dine at a round table, if no women are to sit together, is given by
a) 6! X 5!
b) 30
c) 5! X 4!
d) 7! X 5!

2. The set S = {1,2,3, .....,12} is to be partitioned into three sets A,B,C of equal size.
Thus, A ∪ B ∪ C = S,
A ∩ B = B ∩ C = A ∩ C = φ
The number of ways to partition S is
a) 12!/3!(4!)3
b) 12!/3!(3!)4
c) 12!/(4!)3
d) 12!/(3!)4

• 5. Mathematical Induction - Quiz

1. 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.

2. Let s(k) = 1 + 3 + 5 + ...... + (2k - 1) = 3 + k2. Then which of the following is true?
a) S(1) is correct
b) s(k) ⇒ S(k+1)
c) S(k) ⇒ S(k+1)
d) Principle of mathematical induction can be used to prove the formula.

• 6. Binomial Theorem - Quiz

1. If the expansion in powers of x of the function $\frac{1}{\left (1-ax \right )\left (1-bx \right )}$ is $a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + .....$ then an is
a) $\frac{a^{n} - b^{n}}{b-a}$
b) $\frac{a^{n+1} - b^{n+1}}{b-a}$
c) $\frac{b^{n+1} - a^{n+1}}{b-a}$
d) $\frac{b^{n} - a^{n}}{b-a}$

2. For natural numbers m,n if (1 - y)m(1+y)n = 1 + a1y + a2y2 + .... and a1 = a2 = 10, then (m,n) is
a) (35,20)
b) (45,35)
c) (35,45)
d) (20,45)

• 7. Sequences and Series - Quiz

1. In a geometric progressions consisting of positive terms, each term equals the sum of the next two terms. Then , the common ratio of the progression equals.
a) $\frac{1}{2}\left (1-\sqrt{5} \right )$
b) $\frac{1}{2}\sqrt{5}$
c) $\sqrt{5}$
d) $\frac{1}{2}\left (\sqrt{5} - 1 \right )$

2. The sum of the series $\frac{1}{1.2} - \frac{1}{2.3} + \frac{1}{3.4} - .......$ upto ∞ is equal to
a) 2loge2
b) loge 2-1
c) log2e
d) log2$\left (\frac{4}{e} \right )$

• 8. Limits, Continuity and Differentiabilty - Quiz

1. Suppose the cubic x3 - px + q has three distinct real roots where p > 0 and q > 0 . Then which one of the following holds?
a) The cubic has maxima at both $\sqrt{\frac{p}{3}}$ and -$\sqrt{\frac{p}{3}}$
b) The cubic has minima at $\sqrt{\frac{p}{3}}$ and maxima at -$\sqrt{\frac{p}{3}}$
c) The cubic has minima at -$\sqrt{\frac{p}{3}}$ and maxima at $\sqrt{\frac{p}{3}}$
d) The cubic has minima at both $\sqrt{\frac{p}{3}}$ and -$\sqrt{\frac{p}{3}}$

2. Let f(2) = 4 and f'(2) = 4, Then, $lim_{x \rightarrow 2}\frac{xf\left (2 \right ) - 2f\left (x \right )}{x - 2}$ is given by
a) 2
b) -2
c) -4
d) 3

• 9. Integral Calculas - Quiz

1. Let f(x) be a function satisfying f'(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) g(x) = x2. Then the value of the integral $\int_{0}^{1} f\left (x \right ) g\left (x \right )dx$, is
a) $e - \frac{e^{2}}{2} - \frac{5}{2}$
b) $e\frac{e^{2}}{2} - \frac{3}{2}$
c) $e - \frac{e^{2}}{2} - \frac{3}{2}$
d) $e \frac{e^{2}}{2} \frac{5}{2}$

2. Let f:R → R be a differentiable function having f(2) = 6, f'(2) = $\left (\frac{1}{48} \right)$. Then, $\lim_{x \rightarrow 2}\int_{6}^{f\left (x \right )} \frac{4t^{3}}{x - 2}dx$ dt equals
a) 18
b) 12
c) 36
d) 24

• 10. Differential Equations - Quiz

1. If $\frac{dy}{dx}$ = y + 3 > 0 and y(0) = 2, then y (log 2) is equal to
a) 5
b) 13
c) -2
d) 7

2. The differential equation of the family of circles with fixed radius 5 unit and centre on the line y = 2 is
a) (x - 2)2y'2 = 25 - (y - 2)2
b) (x - 2)y'2 = 25 - (y - 2)2
c) (y - 2)y'2 = 25 - (y - 2)2
d) (y - 2)2y'2 = 25 - (y - 2)2

• 11. Coordinate Geometry - Quiz

1. The point diametricall opposite to the point P(1,0) on the circle x2 + y2 + 2x + 4y - 3 = 0 is
a) (3,4)
b) (3,-4)
c) (-3,4)
d) (-3,-4)

2. In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its ecentricity is
a) $\frac{1}{2}$
b) $\frac{4}{5}$
c) $\frac{1}{\sqrt{5}}$
d) $\frac{3}{5}$

• 12. Three Dimensional Geometry - Quiz

1. If the straight lines $\frac{x - 1}{k}$ = $\frac{y - 2}{k}$ = $\frac{z - 3}{3}$ and $\frac{x - 2}{3}$ = $\frac{y - 3}{k}$ = $\frac{z - 1}{2}$ intersect k is equal to
a) -2
b) -5
c) 5
d) 2

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 $\vec{u}$ = $\hat{i} + \hat{j}, \vec{v}$ = $\vec{i} - \hat{j}$ and $\vec{w}$ = $\hat{i} + 2\hat{j} + 3\hat{k}$ . If $\vec{n}$ is a unit vector such that $\vec{n}$ = 0 and $\vec{v}.\hat{n}$= 0, then $\vec{v}.\hat{n}$ is equal to
a) 0
b) 1
c) 2
d) 3

2. If $\vec{u}, \vec{v}, \vec{w}$ are non-coplanar vectors and p,q are real numbers, then the equality
$\left [3\vec{u} p\vec{v} p\vec{w} \right ] - \left [p\vec{v}\vec{w} q\vec{u} \right ] - \left [2\vec{w} q\vec{v} q\vec{u} \right ]$ holds for
a) exactly two values of (p,q)
b) more than two but not all values of (p,q)
c) all values of (p,q)
d) exactly one value of (p,q)

• 14. Statistics and Probabilty - Quiz

1. If A and B are two mutually exclusive events, then
a) $P\left (A \right ) < P\left (\bar{B} \right )$
b) $P\left (A \right ) > P\left (\bar{B} \right )$
c) P(A) < (B)
d) None of the above

2. The mean of the numbers a,b,8,5 10 is 6 and the varience is 6.80. Then which one of the following gives possible values of a and b?
a) a = 3, b = 4
b) a = 0,b = 7
c) a = 5, b = 2
d) a = 1, b = 6

• 15. Trigonometry - Quiz

1. Let α, β be such that π < α - β < 3π . If sin α + sinβ = $-\frac{21}{65}$ and cos α + cos β = $-\frac{27}{65}$ then the value of $cos\left (\frac{\alpha - \beta}{2} \right )$ is
a) $-\frac{3}{\sqrt{130}}$
b) $\frac{3}{\sqrt{130}}$
c) $\frac{6}{65}$
d) $-\frac{6}{65}$

2. The equation a sinx + b cos x = c where |c| > $\sqrt{a^{2} + b^{2}}$ has
a) a unique solution
b) infinite number of solutions
c) no solution
d) None of the above

• 16. Mathematical Reasoning - Quiz

1. 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.

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)