## Class/Course - Engineering Entrance

### Subject - Mathematics

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

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• 1. Sets, Relations and Functions - Quiz

1. The domain of the function f(x) = $\frac{1}{\sqrt{|x| - x}}$ is
a) (o,∞)
b) (-∞,0)
c) (-∞,∞)-{0}
d) (-∞,∞)

2. The domain of definition of the function
f(x) = $\sqrt{log_{10}\left (\frac{5x - x^{2}}{4} \right )}$ is
a) [1,4]
b) [1,0]
c) [0,5]
d) [5,0]

• 2. Complex Numbers and Quadratic Equations - Quiz

1. If ω is an imaginary cube root of unity, then (1 + ω - ω2)7 equals
a) 128ω
b) -128ω
c) 128ω2
d) -128ω2

2. If ω (≠ 1) is a cube root of unity and (1 + ω)7 = A + Bω . Then , (A,B) equals to
a) (1,1)
b) (1,0)
c) (-1,1)
d) (0,1)

• 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. The number of 3x3 non- singular matrices with four entries as 1 and all other entries and 0, is
a) less than 4
b) 5
c) 6
d) at least 7

• 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. From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then , the middle of such arrangements is
a) at least 500 but less than 750
b) at least 750 but less than 1000
c) at least 1000
d) less than 500

• 5. Mathematical Induction - Quiz

1. 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.

2. 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

• 6. Binomial Theorem - Quiz

1. Let S1 = $\sum_{j=1}^{10}j\left (j-1 \right )^{10}C_{j}, S_{2}$ = $\sum_{j=1}^{10} j^{10}C_{j} \ and \ S_{3}$ = $\sum_{j=1}^{10}$ j2 10Cj
Statement I S3 = 55 × 29.
Statement II S1 = 90 x 28 and S2 = 10 x 28.
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. In the bionomial expansion of (a-b)n, n ≥ 5, the sum of 5th and 6th terms is zero, then $\frac{a}{b}$ equals
a) $\frac{5}{n-4}$
b) $\frac{6}{n-5}$
c) $\frac{n-5}{6}$
d) $\frac{n-4}{5}$

• 7. Sequences and Series - Quiz

1. Then sum of the series
1 + $\frac{1}{4.2!} + \frac{1}{16.4!} + \frac{1}{64.4!} + .... \infty$ is
a) $\frac{e + 1}{2\sqrt{2}}$
b) $\frac{e - 1}{2\sqrt{2}}$
c) $\frac{e + 1}{\sqrt{2}}$
d) $\frac{e - 1}{\sqrt{2}}$

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. For ∈ R, $lim_{x \rightarrow \infty}\left (\frac{x - 3}{x + 2} \right )^{x}$ is equal to
a) e
b) e-1
c) e-5
d) e5

2. The normal to a curve at P(x,y) meets the x-axis at G. If the distance of G from the origin is twice the abscissa of P, then the curve is a
a) ellipse
b) parabola
c) circle
d) hyperbola

• 9. Integral Calculas - Quiz

1. $\int_{-\pi}^{\pi}\frac{2x\left (1 + sinx \right )}{1 + cos^{2}x}dx$ is
a) $\frac{\pi^{2}}{4}$
b) π2
c) 0
d) $\frac{\pi}{2}$

2. The value of $\int_{0}^{1}\frac{8 log\left (1 + x \right )}{1 + x^{2}}dx$ is
a) $\frac{\pi}{8}log2$
b) $\frac{\pi}{2}log2$
c) log 2
d) π log 2

• 10. Differential Equations - Quiz

1. The order and degree of the differential equation $\left (1 + 3\frac{dy}{dx} \right )^{\frac{2}{3}}$ = $4\frac{d^{3}y}{dx^{3}}$ are
a) $\left (1, \frac{2}{3} \right )$
b) (3,1)
c) (3,3)
d) (1,2)

2. The differential equation of all circles passing through the origin and having their centres on the x - axis is
a) $x^{2}$ = $y^{2} + xy\frac{dy}{dx}$
b) $x^{2}$ = $y^{2} + 3xy\frac{dy}{dx}$
c) $x^{2}$ = $y^{2} + 2xy\frac{dy}{dx}$
d) $x^{2}$ = $y^{2} - 2xy\frac{dy}{dx}$

• 11. Coordinate Geometry - Quiz

1. The equation of the ellipse whose foci are (±2,0) and eccentricity is 1/2, is
a) $\frac{x^{2}}{12} + \frac{y^{2}}{16}$ = 1
b) $\frac{x^{2}}{16} + \frac{y^{2}}{12}$ = 1
c) $\frac{x^{2}}{16} + \frac{y^{2}}{8}$ = 1
d) None of these

2. The eccentricity of an ellipse with its centre at the origin, is $\frac{1}{2}$ . If one of the direction is x = 4, then the equilibrium of the ellipse is
a) 3x2 + 4y2 = 1
b) 3x2 + 4y2 = 12
c) 4x2 + 3y2 = 12
d) 4x2 + 3y2 = 1

• 12. Three Dimensional Geometry - Quiz

1. If the angle between the line x =$\frac{y-1}{2}$=$\frac{z-3}{3}$ and the plane x+2y+3z= 4 is $cos^{-1}\left ( \sqrt{\frac{5}{14}} \right )$ then λ equals to
a) $\frac{3}{2}$
b) $\frac{2}{5}$
c) $\frac{5}{3}$
d) $\frac{2}{3}$

2. A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The coordinates of each of the points of intersection are given by
a) (3a, 3a, 3a), (a, a, a)
b) (3a, 2a, 3a), (a, a, a)
c) (3a, 2a, 3a), (a, a, 2a)
d) (2a, 3a, 3a),(2a, a, a)

• 13. Vector Algebra - Quiz

1. The value of a, for which the points , A, B, C with position vectors $2\vec{i} - \vec{j} + \vec{k}, \vec{i} - 3\vec{j} - 5\vec{k}$ and $\vec{a} - 3\vec{j} + \vec{k}$ respectively are the vertices of a right angled triangle with C = $\frac{\pi }{2}$ are
a) -2 and -1
b) -2 and 1
c) 2 and -1
d) 2 and 1

2. Let $\vec{a}$ = $\hat{i} + \hat{j} + \hat{k} , \vec{b}$ = $\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{c}$ and $x\hat{i} + \left (x - 2 \right )\hat{j} - \hat{k}$ . If the vector $\vec{c}$ lies in the plane of $\vec{a} and \vec{b}$ , then x equals
a) 0
b) 1
c) -4
d) -2

• 14. Statistics and Probabilty - Quiz

1. A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event the number obtained is less than S. Then P(A ∪ B) is
a) $\frac{2}{5}$
b) $\frac{3}{5}$
c) 0
d) 1

2. 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

• 15. Trigonometry - Quiz

1. Let A and B denote the statements
A : cosα + cosβ + cosγ = 0
B : sin α + sinβ + sinγ = 0
If cos(β - y) + cos( γ - α) + cos(α - β) = $-\frac{3}{2}$ , then
a) A is true and B is false
b) A is false and B is true
c) both A and B are true

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. 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. 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)