JEE MAIN TEST – 8 – JEE MATH APEX

Mathematics Questions (Continued)

1. If the function $f(x) = c x e^{-x} - \frac{x^2}{2} + x$ is decreasing for every $x \in (-\infty, 0]$ , then the least value of $c^2$ is equal to

(A) 1

(B) 2

(D) 4

Click to View Answer

Correct Answer: [Insert Correct Option]

2. The $x$ -intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point $(1,1)$ is

(A) $x e^{\frac{x}{y}} = e$

(B) $x e^{\frac{y}{x}} = e$

(D) $y e^{\frac{x}{y}} = e$

Click to View Answer

Correct Answer: [Insert Correct Option]

3. Assertion (A): If $R$ is a relation defined on set of natural numbers $N$ such that $R = \{(x, y) : x, y \in N \text{ and } x + y = 24\}$ then $R$ is an equivalence relation.
Reason (R): A relation is said to be an equivalence relation if it is reflexive, symmetric and transitive.

(A) A is True, R is True; R is a correct explanation for A.

(B) A is True, R is True; R is not a correct explanation for A.

(D) A is False, R is True.

Click to View Answer

Correct Answer: [Insert Correct Option]

4. Let $f(x) = 12\left(\frac{e^{3x} - 3e^x}{e^{2x} - 1}\right)$ be defined for $x > 0$ and $g(x)$ be the inverse of $f(x)$ . If $\int_g^{27} g(x) \, dx = a \ln 3 - b \ln 2 - c$ then the value of $(a - (b + c))$ is

(A) 7

(B) 6

(D) 71

Click to View Answer

Correct Answer: [Insert Correct Option]

5. Let $f(x)$ be a non constant twice differentiable function on $\mathbb{R}$ such that $f(2 + x) = f(2 - x)$ and $f'\left(\frac{1}{2}\right) = f'(1) = 0$ . Then minimum number of root(s) of equation $f''(x) = 0$ in $(0, 4)$ is/are

(A) 2

(B) 4

(D) 6

Click to View Answer

Correct Answer: [Insert Correct Option]

6. A circle of radius 2 units having centre in fourth quadrant passes through the vertex and focus of parabola $y^2 = 4x$ and touches the parabola $y = -\left(x - \frac{1}{2}\right)^2 - \alpha, \alpha > 0$ then the value of $(2\alpha - 4)^2$ is

(A) 15

(B) 18

(D) 25

Click to View Answer

Correct Answer: [Insert Correct Option]

7. Value of $\int_0^{\frac{\pi}{2}} \frac{\sin 8x}{\sin x} \, dx$ is

(A) $\frac{152}{105}$

(B) $\frac{52}{105}$

(D) $\frac{152}{35}$

Click to View Answer

Correct Answer: [Insert Correct Option]

8. If the line $x - 1 = 0$ divides the area bounded by the curves $2x + 1 = \sqrt{4y + 1}$ , $y = x$ and $y = 2$ in two regions of area $A_1$ and $A_2$ ( $A_1 < A_2$ ) then $\left(A_1^{-2} - A_2^{-2}\right)$ is equal to

(A) 4

(B) 5

(D) 8

Click to View Answer

Correct Answer: [Insert Correct Option]

9. Match List-I with List-II:

List-I	List-II
(A) Let $a_1, a_2, a_3, \ldots$ be AP. If $\sum_{r=1}^{\infty} \frac{a_r}{2^r} = 4$ then $4a_2$ is equal to	(I) 16
(B) $(20)^{19} + 2(21)(20)^{18} + 3(21)^2(20)^{17} + \ldots + 20(21)^{19} = K(20)^{19}$ , then $\frac{K}{100}$ is equal to	(II) 4
(C) If number of integral solution to the equation $x \geq 1, y \geq 3, z \geq 4$ is $K$ then $\frac{K + 7}{7}$ is	(III) 15
(D) Let $\frac{1}{16}, a$ and $b$ be in GP and $\frac{1}{a}, \frac{1}{b}, 6$ be in AP where $a, b > 0$ then $72(a + b)$ is	(IV) 14

(A) (A) – II, (B) – IV, (C) – IV, (D) – I

(B) (A) – I, (B) – II, (C) – I, (D) – IV

(D) (A) – IV, (B) – II, (C) – I, (D) – II

Click to View Answer

Correct Answer: [Insert Correct Option]

10. A game board is shown in the diagram below.

[Diagram: Linear Game Track with numbered squares 1-10 on top and 18-11 on bottom]

Player take turns to roll an ordinary die, then move their counter forward from ‘START’ a number of squares equal to the number rolled with the die. If a player’s counter ends its move on a cross marked square, then it is moved back to START. Let $\alpha$ denotes the probability that player’s counter is on START after rolling the die twice and let $\beta$ denotes the probability that after rolling the die thrice, a player’s counter is on square numbered 17, then the value of $\frac{\alpha}{\beta}$ is

(A) 24

(B) 21

(D) 18

Click to View Answer

Correct Answer: [Insert Correct Option]

11. If $\left(\sin^{-1} a\right)^2 + \left(\cos^{-1} b\right)^2 + \left(\sec^{-1} c\right)^2 + \left(\csc^{-1} d\right)^2 = \frac{5\pi^2}{2}$ then the value of $\left(\sin^{-1} a\right)^2 - \left(\cos^{-1} b\right)^2 + \left(\sec^{-1} c\right)^2 - \left(\csc^{-1} d\right)^2$ is

(A) $-\pi^2$

(B) $-\frac{\pi^2}{2}$

(D) $\frac{\pi^2}{2}$

Click to View Answer

Correct Answer: [Insert Correct Option]

12. Tangents are drawn from the point $(\alpha, \beta)$ to the hyperbola $3x^2 - 2y^2 = 6$ and are inclined at angles $\theta$ and $\phi$ to the $x$ -axis. If $\tan \theta \tan \phi = 2$ then value of $2\alpha^2 - \beta^2$ is

(A) 7

(B) -7

(D) -1

Click to View Answer

Correct Answer: [Insert Correct Option]

13. Let $\frac{\cot 3^\circ}{\cot^2 3^\circ - 3} + \frac{3 \cot 9^\circ}{\cot^2 9^\circ - 3} + \frac{9 \cot 27^\circ}{\cot^2 27^\circ - 3} + \frac{27 \cot 81^\circ}{\cot^2 81^\circ - 3} = x \cot 27^\circ + y \cot 87^\circ$ . Then value of $4(x + y)$ is

(A) 40

(B) 42

(D) 84

Click to View Answer

Correct Answer: [Insert Correct Option]

14. Let $P(x)$ be polynomial $x^3 + ax^2 + bx + c$ where $a, b, c \in \mathbb{R}$ . If $P(-3) = P(2) = 0$ and $P'(-3) < 0$ . Which of the following is a possible value of $c$ .

(A) -27

(B) -18

(D) -3

Click to View Answer

Correct Answer: [Insert Correct Option]

15. Statement-I: Let $B$ be matrix of order $3 \times 3$ and $\text{adj } B = A$ . If $M$ and $N$ are matrices of order $3 \times 3$ such that $\det(M) = 1 = \det(N)$ then $\text{adj}(N^{-1} B M^{-1}) = MAN$ . (where $\det(X)$ denotes determinant of matrix $X$ ; $\text{adj}(Y)$ denotes adjoint of matrix $Y$ )
Statement-II: If $P$ is non-singular square matrix of order $3 \times 3$ then $\text{adj}(P^{-1}) = (\text{adj } P)^{-1}$

(A) Statement-I is true, Statement-II is false.

(B) Statement-I is false, Statement-II is true.

(D) Statement-I is false, Statement-II is false.

Click to View Answer

Correct Answer: [Insert Correct Option]

16. Let $f(x) = (x^2 - 9) |x^3 - 6x^2 + 11x - 6| + \frac{x}{1 + |x|}$ and $g: (-2, 2) \to \mathbb{R}, g(x) = [x] |x^2 - 1| + \sin \left( \frac{\pi}{[x] + 3} \right) - [x + 1]$ . If ‘ $m$ ‘ denotes number of points where $f(x)$ is not differentiable and ‘ $n$ ‘ denotes the number of points where $g(x)$ is discontinuous then $(m + n)$ is (where $[.]$ denotes the Greatest Integer Function)

(A) 3

(B) 2

(D) 6

Click to View Answer

Correct Answer: [Insert Correct Option]

17. Equation of plane which passes through the point of intersection of the lines $\frac{x-1}{3} = \frac{y-2}{1} = \frac{z-3}{2}$ and $\frac{x-3}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ and has the largest distance from the origin is $ax + by + cz + 50 = 0$ then $|a + b + c|$ is

(A) 12

(B) 7

(D) 5

Click to View Answer

Correct Answer: [Insert Correct Option]

18. Let $f(x)$ be a real valued function such that $f(x) = \cos x + \int_{-\pi}^\pi (\cos x + |u| f(u)) \, du$ . If $M$ and $m$ are the maximum and minimum values of the function $f(x)$ respectively then $\frac{M}{m}$ is

(A) $\frac{2 - \pi}{6 + \pi}$

(B) $-\frac{(\pi + 1)}{(\pi + 6)}$

(D) $\frac{(2\pi)}{(3 - \pi)}$

Click to View Answer

Correct Answer: [Insert Correct Option]

19. An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $\mu$ is the average marks of girls and $\sigma^2$ is the variance of marks of all 50 candidates, then $\mu + \sigma^2$ is equal to

(A) 25

(B) 20

(D) 30

Click to View Answer

Correct Answer: [Insert Correct Option]

20. Let $\alpha$ be the root of $x^2 + x + 1 = 0$ . For some values of ‘ $n$ ‘, if $\left(1 - \alpha + \alpha^2\right) \left(1 - \alpha^2 + \alpha^4\right) \left(1 - \alpha^3 + \alpha^6\right) \left(1 - \alpha^4 + \alpha^8\right) \ldots \left(1 - \alpha^n + \alpha^{2n}\right) = a^b$ where $(a, b)$ is orthocentre of triangle with $(2, 12), (5, 12 + \sqrt{3})$ and $(3, 12 - \sqrt{3})$ as co-ordinates of its vertices. Then sum of possible values of $n$ is

(A) 33

(B) 15

(D) 23

Click to View Answer

Correct Answer: [Insert Correct Option]

21. Let $\mathbf{X}$ and $\mathbf{Y}$ be the set of words which can be formed using all the letters of the words SHREYANSH and SANIDHYA respectively. A set is randomly chosen and a word is selected at random from it. If the probability that it contains at least one pair of alike letters together is $\frac{p}{q}$ (where $p$ and $q$ are coprime) then the value of $(q - 3p)$ is

Click to View Answer

Correct Answer: [Insert Correct Answer]

22. Let $\vec{a}, \vec{b}, \vec{c}$ be the vectors representing three coterminous edges of tetrahedron such that $\vec{a} \wedge \vec{b} = \vec{b} \wedge \vec{c} = \vec{c} \wedge \vec{a} = \frac{\pi}{3}$ and $4 \vec{a} \cdot \vec{a} + 3 \vec{b} \cdot \vec{b} + 2 \vec{c} \cdot \vec{c} = 144$ . If $V$ is volume of the tetrahedron, then the maximum value of $V$ is (where $\vec{a} \wedge \vec{b}$ represent angle between $\vec{a}$ and $\vec{b}$ )

Click to View Answer

Correct Answer: [Insert Correct Answer]

23. If the complete set of values of ‘ $a$ ‘ for which the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2 \sin 2x - 3 \cos^2 x - (a^2 + a - 7)x + 5, a \in \mathbb{R}$ is strictly increasing is $[p, q]$ (where $p, q$ are integers) then $|p + q|$ is

Click to View Answer

Correct Answer: [Insert Correct Answer]

24. Let $755\ldots57$ denotes $(r + 2)$ digit number where the first and last digit are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \ldots + 75\ldots57$ . If $S = \frac{68 \cdot 10^\lambda + 11020}{81}$ , then $\lambda$

Click to View Answer

Correct Answer: [Insert Correct Answer]

25. A point $P(x, y)$ moves in xy plane in such a way that $\sqrt{2} \leq |x + y| + |x - y| \leq 3\sqrt{2}$ . Area of region representing all possible of point $P$ is equal to

Click to View Answer

Correct Answer: [Insert Correct Answer]