2026

AISSCE(Board Exam)

Full Paper Analysis & Solutions

Mathematics

Multiple Choice Questions

1 REMEMBER MCQ_SINGLE 1 Mark(s) 2026
#1488
For any square matrix A with real entries, if $A + A'$ is a symmetric matrix then :
(A)
(A - A') cannot be a skew symmetric matrix
(B)
(A - A') is a skew symmetric matrix
(C)
A is always a symmetric matrix
(D)
A is always a skew symmetric matrix
EASY
2 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2026
#1489
A matrix B = $[b_{ij}]_{m \times m}$ is said to be a diagonal matrix, if :
(A)
$b_{ij} = 0$, when $i = j$
(B)
$b_{ij} = 1$, when $i = j$
(C)
$b_{ij} = 1$, when $i \neq j$
(D)
$b_{ij} = 0$, when $i \neq j$
EASY
3 REMEMBER MCQ_SINGLE 1 Mark(s) 2026
#1493
The general solution for the differential equation $\frac{dy}{dx} = e^{3x-y}$ is
(A)
$3e^y = e^{3x} + C$
(B)
$\log (3x - y) = C$
(C)
$e^{3x-y} = C$
(D)
$-e^y + 3e^{3x} = C$
MEDIUM
4 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1670
A relation R on set $A=\{1,2,3\}$ defined as $R=\{(1,1), (2,2), (1,2)\}$ is
(A)
Reflexive only
(B)
Reflexive and Transitive
(C)
Symmetric and Transitive
(D)
Transitive only
MEDIUM
5 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1671
The principal value of $\sec^{-1}(\sqrt{2})+2\text{cosec}^{-1}(-\sqrt{2})$ is:
(A)
$-\frac{\pi}{2}$
(B)
$-\frac{\pi}{4}$
(C)
$\frac{\pi}{4}$
(D)
$\frac{\pi}{2}$
MEDIUM
6 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1672
For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined?
(A)
$\tan^{-1}:R\rightarrow(-\frac{\pi}{2},\frac{\pi}{2})$
(B)
$\sec^{-1}:R-(-1,1)\rightarrow[0,\pi]-\{\frac{\pi}{2}\}$
(C)
$\cot^{-1}:R\rightarrow(0,\pi)$
(D)
$\text{cosec}^{-1}:R-(-1,1)\rightarrow[-\frac{\pi}{2},\frac{\pi}{2}]$
MEDIUM
7 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1673
The following graph represents:
(A)
$y=\cos^{-1}x$
(B)
$y=\sec^{-1}x$
(C)
$y=\tan^{-1}x$
(D)
$y=\text{cosec}^{-1}x$
MEDIUM
8 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1674
The domain of $f(x)=\cos^{-1}(2x-5)$ is:
(A)
[-1,1]
(B)
[4, 6]
(C)
$[-7,-3]$
(D)
[2, 3]
MEDIUM
9 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1675
If a square matrix A is such that $A^{2}=A$ and $(I-A)^{3}=xA+I$ then value of x must be:
(A)
7
(B)
5
(C)
-7
(D)
-1
MEDIUM
10 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1676
If $A=\begin{bmatrix}1&a&b\\-1&2&c\\0&5&3\end{bmatrix}$ is a symmetric matrix, then the value of $3a+b+c$ is
(A)
2
(B)
6
(C)
4
(D)
0
MEDIUM
11 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1677
If $A^{2}=4A+3I$ and $A^{-1}=xA+yI$, then the value of $(x+y)$ is :
(A)
-1
(B)
1
(C)
$\frac{5}{3}$
(D)
7
MEDIUM
12 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1678
If $\begin{vmatrix}-1&-2&5\\-2&a&-1\\0&4&2a\end{vmatrix}=-86$, then the sum of all possible values of a is
(A)
4
(B)
5
(C)
-4
(D)
9
MEDIUM
13 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1679
If $adj(B)=\begin{bmatrix}\frac{1}{3}&0&0\\0&\frac{1}{3}&0\\0&0&\frac{1}{3}\end{bmatrix}$, then the value of det $(B^{-1})=$
(A)
$\frac{1}{3}$
(B)
$\frac{1}{9}$
(C)
3
(D)
9
MEDIUM
14 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1680
If $A=[a_{ij}]$ is a $2\times 2$ matrix whose elements are given by $a_{ij}=\frac{|i-3j|}{2}$, then $A^{\prime}$ is:
(A)
$\begin{bmatrix}1&\frac{5}{2}\\\frac{1}{2}&2\end{bmatrix}$
(B)
$\begin{bmatrix}1&\frac{1}{2}\\\frac{5}{2}&2\end{bmatrix}$
(C)
$\begin{bmatrix}2&\frac{5}{2}\\\frac{1}{2}&1\end{bmatrix}$
(D)
$\begin{bmatrix}2&\frac{1}{2}\\\frac{5}{2}&1\end{bmatrix}$
MEDIUM
15 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1681
Let $A=\begin{bmatrix}0&-3&4\\1&0&2\end{bmatrix}$ and $B=\begin{bmatrix}-3&0&1\\2&4&0\end{bmatrix}$. If $A + B + C = O$, then matrix C is:
(A)
$\begin{bmatrix}-3&-3&5\\3&4&2\end{bmatrix}$
(B)
$\begin{bmatrix}3&3&-5\\-3&-4&-2\end{bmatrix}$
(C)
$\begin{bmatrix}3&3&5\\-3&-4&-2\end{bmatrix}$
(D)
$\begin{bmatrix}-3&-3&-5\\3&4&2\end{bmatrix}$
EASY
16 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1682
If $A=\begin{bmatrix}\cos x&-\sin x\\\sin x&\cos x\end{bmatrix}$ and $A+A^{\prime}=I$, then the value of $x \in [0,\frac{\pi}{2}]$ is
(A)
$0$
(B)
$\frac{\pi}{3}$
(C)
$\frac{\pi}{4}$
(D)
$\frac{\pi}{2}$
MEDIUM
17 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1683
For a square matrix A, $(3A)^{-1}=$
(A)
$3A^{-1}$
(B)
$9A^{-1}$
(C)
$\frac{1}{3}A^{-1}$
(D)
$\frac{1}{9}A^{-1}$
MEDIUM
18 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1684
If matrix $A=\begin{bmatrix}-p&q\\r&p\end{bmatrix}$ is such that $A^{2}=I$ then :
(A)
$1+p^{2}+qr=0$
(B)
$1-p^{2}-qr=0$
(C)
$1-p^{2}+qr=0$
(D)
$1+p^{2}-qr=0$
MEDIUM
19 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1685
If $\begin{bmatrix}4\\1\\3\end{bmatrix}A=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}$, then order of A must be:
(A)
$3\times1$
(B)
$1\times3$
(C)
$1\times1$
(D)
$3\times3$
MEDIUM
20 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1686
If A is a non-singular matrix, then which of the following is not true?
(A)
adj A is singular
(B)
$(adj~A)^{-1}=adj(A^{-1})$
(C)
$|A|\ne 0$
(D)
$A^{-1}$ exists
MEDIUM
21 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1687
If A is a square matrix such that $A^{2}=A$ then $(A-I)^{3}-A$ is equal to :
(A)
I
(B)
$-I$
(C)
A
(D)
$A^{2}$
MEDIUM
22 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1688
If the area of $\Delta$ ABC with vertices $A(3,1)$, $B(-2,1)$ and $C(0,k)$ is 5 sq. units, then values of k are:
(A)
3, 1
(B)
-1, 3
(C)
-1, 2
(D)
0, 2
MEDIUM
23 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1689
If A and B are skew-symmetric matrices of same order, then $AB^{\prime}+BA^{\prime}$ is a/an:
(A)
symmetric matrix
(B)
skew-symmetric matrix
(C)
null matrix
(D)
identity matrix
MEDIUM
24 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1690
If A and B are square matrices of same order, then which of the following statements is/are always true? (i) $(A+B)(A-B)=A^{2}-B^{2}$ (ii) $AB=BA$ (iii) $(A+B)^{2}=A^{2}+AB+BA+B^{2}$ (iv) $AB=0 \Rightarrow A=0$ or $B=0$
(A)
Only (i) and (iii)
(B)
Only (ii) and (iii)
(C)
Only (iii)
(D)
Only (iii) and (iv)
MEDIUM
25 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1691
If $f(x)=\begin{cases}\frac{x^{2}-4x-5}{x+1},&x\ne-1\\k,&x=-1\end{cases}$ is continuous at $x=-1$ then the value of k is:
(A)
Any real value
(B)
6
(C)
-1
(D)
-6
MEDIUM
26 COMPETENCY EVALUATE MCQ_SINGLE 1 Mark(s) 2026
#1692
The value of k for which the function $f(x)=\begin{cases} x^{2}\sin\frac{1}{x}, & x\ne0 \\ k(x+1), & x=0 \end{cases}$ is a continuous function, is:
(A)
$\frac{1}{4}$
(B)
2
(C)
$\frac{1}{2}$
(D)
0
MEDIUM
27 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1693
If $f(x)=\begin{cases}\frac{\sin x}{x}+\cos x, & x\ne0 \\ k, & x=0\end{cases}$ is continuous at $x=0$, then the value of k is:
(A)
$0$
(B)
-2
(C)
-1
(D)
2
MEDIUM
28 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1694
The greatest integer function, $f(x)=[x]$ for $0
(A)
At only one point
(B)
At only two points
(C)
At no point
(D)
At three points
MEDIUM
29 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1695
Differential of $e^{e^{x}}$ with respect to x is:
(A)
$\log x$
(B)
$e^{e^{x}}$
(C)
$e^{x}e^{e^{x}}$
(D)
$(e^{x})^{2}$
EASY
30 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1696
Derivative of $\cos^{-1}(\frac{\sin x+\cos x}{\sqrt{2}})$, $-\frac{\pi}{4}
(A)
$-1$
(B)
1
(C)
$\frac{\pi}{4}$
(D)
$-\frac{\pi}{4}$
MEDIUM
31 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1697
If $\sin^{-1}x=y$ then $\frac{dy}{dx}$ is:
(A)
$\cos^{-1}x$
(B)
$\cos y$
(C)
$\frac{1}{1-x^{2}}$
(D)
$\sec y$
MEDIUM
32 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1698
If $e^{-x}+e^{-y}=2$, then $\frac{dy}{dx}$ is
(A)
$e^{x-y}$
(B)
$e^{y-x}$
(C)
$-e^{x-y}$
(D)
$-e^{y-x}$
MEDIUM
33 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1699
Absolute minimum value of $f(x)=(x-2)^{2}+5$ in the interval [-3, 2] is :
(A)
-3
(B)
2
(C)
5
(D)
30
MEDIUM
34 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1700
The surface area of a sphere when its volume changes at the same rate as its radius is :
(A)
$4\pi$ sq. units
(B)
1 sq. unit
(C)
4 sq. units
(D)
$\pi$ sq. units
MEDIUM
35 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1701
The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is:
(A)
$400\pi~cm^{3}/cm$
(B)
$100\pi~cm^{3}/cm$
(C)
$50\pi~cm^{3}/cm$
(D)
$25\pi~cm^{3}/cm$
MEDIUM
36 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1702
For $f(x)=x+\frac{1}{x} (x \ne 0)$
(A)
local maximum value is 2
(B)
local minimum value is -2
(C)
local maximum value is -2
(D)
local minimum value < local maximum value
MEDIUM
37 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1703
$\int\frac{dx}{2^{x}+2^{-x}}$ is equal to:
(A)
$\tan^{-1}(2^{x})+C$
(B)
$\tan^{-1}(2^{-x})+C$
(C)
$\frac{\tan^{-1}(2^{x})}{\log 2}+C$
(D)
$(\log 2)\tan^{-1}(2^{x})+C$
MEDIUM
38 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1704
$\int\frac{dx}{\sqrt{25-16x^{2}}}$ is equal to:
(A)
$\frac{1}{5}\sin^{-1}4x+C$
(B)
$\frac{1}{25}\sin^{-1}16x+C$
(C)
$\frac{1}{4}\sin^{-1}\frac{4x}{5}+C$
(D)
$\frac{1}{16}\sin^{-1}\frac{4x}{5}+C$
MEDIUM
39 REMEMBER MCQ_SINGLE 1 Mark(s) 2026
#1705
$\int\frac{1}{\sqrt{1+\cos 2x}}dx$ is equal to:
(A)
$\log|\cos x|+C$
(B)
$\frac{1}{\sqrt{2}}\log|\sec x-\tan x|+C$
(C)
$\frac{1}{\sqrt{2}}\log|\sec x+\tan x|+C$
(D)
$\log|\sin 2x|+C$
EASY
40 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1706
If $\int_{0}^{1}\frac{dx}{e^{x}+e^{-x}}=\tan^{-1}e+k$, then the value of k is:
(A)
e
(B)
$\frac{\pi}{4}$
(C)
$0$
(D)
$-\frac{\pi}{4}$
MEDIUM
41 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1707
$\int_{-1}^{1}(1-|x|)dx$ is equal to:
(A)
$2\int_{0}^{1}(1+x)dx$
(B)
$0$
(C)
$2\int_{-1}^{0}(1+x)dx$
(D)
$2\int_{-1}^{0}(1-x)dx$
MEDIUM
42 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1708
If $\int_{0}^{2a}\frac{1}{1+4x^{2}}dx=\frac{\pi}{6}$, then the value of a is
(A)
$\frac{\sqrt{3}}{4}$
(B)
$-\frac{\sqrt{3}}{4}$
(C)
$\sqrt{3}$
(D)
$2\sqrt{3}$
MEDIUM
43 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1709
The value of $\int_{-5}^{-1}\frac{1}{x}dx$ is equal to:
(A)
$-\log 5$
(B)
$x^{6}$
(C)
$\log(-5)$
(D)
$x^{-6}$
EASY
44 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1710
Which of the following expressions will give the area of region bounded by the curve $y=x^{2}$ and line $y=16$?
(A)
$\int_{0}^{4}x^{2}dx$
(B)
$2\int_{0}^{4}x^{2}dx$
(C)
$\int_{0}^{16}\sqrt{y}dy$
(D)
$2\int_{0}^{16}\sqrt{y}dy$
MEDIUM
45 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1711
The area of the shaded region of the circle given below is equal to :
(A)
$\int_{1}^{3}\sqrt{9-y^{2}}dy$
(B)
$2\int_{1}^{3}\sqrt{9-y^{2}}dy$
(C)
$\int_{0}^{3}\sqrt{9-x^{2}}dx$
(D)
$2\int_{0}^{3}\sqrt{9-x^{2}}dx$
MEDIUM
46 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1712
The area of the region bounded by the curve $y=x$ and x-axis, between $x=0$ and $x=2$ is:
(A)
2 sq. units
(B)
$\frac{1}{2}$ sq. unit
(C)
1 sq. unit
(D)
4 sq. units
EASY
47 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1713
An ant is observed crawling on a sheet of paper along a straight line given by equation $y=2x-4$. Area of the surface covered by the ant bounded by y-axis, x-axis and $x=1$ is :
(A)
1 sq. unit
(B)
3 sq. units
(C)
2 sq. units
(D)
4 sq. units
MEDIUM
48 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1714
Product of the order and degree of differential equation $1+(\frac{dy}{dx})^{3}=\lambda(\frac{d^{3}y}{dx^{3}})^{2}$ is:
(A)
5
(B)
6
(C)
2
(D)
3
MEDIUM
49 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1715
The general solution of the differential equation $\frac{dy}{dx}=\frac{\sqrt{y}}{\sqrt{x}}$ is
(A)
$\log\sqrt{y}=\log\sqrt{x}+C$
(B)
$\sqrt{y}+\sqrt{x}=C$
(C)
$\sqrt{y}-\sqrt{x}=C$
(D)
$\log\sqrt{y}+\log\sqrt{x}=C$
MEDIUM
50 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1716
The integrating factor of the differential equation $2x\frac{dy}{dx}-y=3$ is
(A)
$\sqrt{x}$
(B)
$\frac{1}{\sqrt{x}}$
(C)
$e^{x}$
(D)
$e^{-x}$
MEDIUM
51 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1717
The general solution for the differential equation $\frac{dy}{dx}=e^{3x-y}$ is:
(A)
$3e^{y}=e^{3x}+C$
(B)
$\log(3x-y)=C$
(C)
$e^{3x-y}=C$
(D)
$-e^{y}+3e^{3x}=C$
MEDIUM
52 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1718
Which of the following is not a Linear Differential Equation?
(A)
$(1+x^{2})dy+2xy~dx=\cot x~dx$
(B)
$y+\frac{d}{dx}(xy)=x(\sin x+\log x)$
(C)
$x(1+y^{2})dx-y(1+x^{2})dy=0$
(D)
$y~dx-(x+3y^{2})dy=0$
MEDIUM
53 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1719
$\frac{dy}{dx}=F(x,y)$ will be a homogeneous differential equation for which of the following functions? (i) $F(x,y)=3x+2y$ (ii) $F(x,y)=\sin\frac{y}{x}+\log y-\log x$ (iii) $F(x,y)=e^{y/x}+1$ (iv) $F(x,y)=\sqrt{x^{2}+y^{2}}-y$
(A)
(i) and (ii)
(B)
(i), (ii) and (iii)
(C)
(ii), (iii) and (iv)
(D)
(ii) and (iii)
MEDIUM
54 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1720
The order and degree of the differential equation $1+(\frac{d^{3}y}{dx^{3}})^{3}=\lambda\frac{d^{2}y}{dx^{2}}$ is:
(A)
Order = 3, Degree = 3
(B)
Order = 2, Degree = 2
(C)
Order = 3, Degree = 1
(D)
Order = 2, Degree = 1
MEDIUM
55 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1721
If vectors $\vec{a}=3\hat{i}+2\hat{j}+\lambda\hat{k}$ and $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$, represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of $\lambda$ is :
(A)
1
(B)
$\frac{5}{2}$
(C)
$\frac{2}{5}$
(D)
0
MEDIUM
56 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1722
If $|\vec{a}|=5$ and $-2 \le \lambda \le 1$ then the sum of greatest and the smallest value of $|\lambda\vec{a}|$ is
(A)
-5
(B)
5
(C)
10
(D)
15
MEDIUM
57 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1723
If position vector $\vec{p}$ of a point (24, n) is such that $|\vec{p}|=25$, then the value of n is:
(A)
$\pm 49$
(B)
$\pm 5$
(C)
$\pm 1$
(D)
$\pm 7$
EASY
58 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1724
Three points $A(0,1,1)$, $B(2,0,-1)$ and $C(1,0,3)$ form $\Delta ABC$. The ar ($\Delta ABC$) is:
(A)
$\frac{\sqrt{53}}{2}$ sq. units
(B)
$\sqrt{53}$ sq. units
(C)
$\frac{\sqrt{11}}{2}$ sq. units
(D)
$\sqrt{11}$ sq. units
MEDIUM
59 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1725
Vector of magnitude 3 making equal angles with x and y axes and perpendicular to z axis is
(A)
$\hat{i}+2\sqrt{2}\hat{j}$
(B)
$3\hat{k}$
(C)
$\frac{3\sqrt{2}}{2}\hat{i}+\frac{3\sqrt{2}}{2}\hat{j}$
(D)
$\sqrt{3}\hat{i}+\sqrt{3}\hat{j}+\sqrt{3}\hat{k}$
MEDIUM
60 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1726
For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true ?
(A)
$\vec{a}.\vec{b}\le|\vec{a}||\vec{b}|$
(B)
$|\vec{a}+\vec{b}|\ge|\vec{a}|+|\vec{b}|$
(C)
$|\vec{a}-\vec{b}|=|\vec{a}|-|\vec{b}|$
(D)
$|\vec{a}\times\vec{b}|\ge|\vec{a}||\vec{b}|$
MEDIUM
61 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1727
If $(\vec{a}+\vec{b}).(\vec{a}-\vec{b})=198$ and $|\vec{a}|=10|\vec{b}|$, then :
(A)
$|\vec{a}|=\sqrt{2}$
(B)
$|\vec{b}|=\sqrt{2}$
(C)
$|\vec{b}|=10\sqrt{2}$
(D)
$|\vec{a}|=\frac{10}{\sqrt{2}}$
MEDIUM
62 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1728
If $(3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}$ then the values of p and q are :
(A)
$p=-\frac{2}{3}, q=\frac{5}{3}$
(B)
$p=-\frac{8}{3}, q=\frac{20}{3}$
(C)
$p=\frac{20}{3}, q=-\frac{8}{3}$
(D)
$p=0, q=0$
MEDIUM
63 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1729
If points $(2, 3)$, $(0, 4)$ and $(p, 2)$ are collinear, then the value of p is:
(A)
$\frac{4}{7}$
(B)
$-\frac{3}{7}$
(C)
4
(D)
-4
MEDIUM
64 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1730
If $l_{1}$, $m_{1}$, $n_{1}$ and $l_{2}$, $m_{2}$, $n_{2}$ are direction cosines of lines $L_{1}$ and $L_{2}$ respectively and $\theta$ is the acute angle between them, then :
(A)
$\cos\theta=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}$
(B)
$\sin\theta=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}$
(C)
$\tan\theta=\frac{l_{1}}{l_{2}}+\frac{m_{1}}{m_{2}}+\frac{n_{1}}{n_{2}}$
(D)
$\cos\theta=|l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}|$
EASY
65 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1731
Direction cosines of the line given by equations $\frac{2x-1}{4}=\frac{1-y}{3}=\frac{-z}{6}$ are
(A)
$2,-3,-6$
(B)
$\frac{2}{7},\frac{-3}{7},\frac{-6}{7}$
(C)
$\frac{2}{7},\frac{-3}{7},\frac{6}{7}$
(D)
$\frac{4}{\sqrt{61}},\frac{-3}{\sqrt{61}},\frac{-6}{\sqrt{61}}$
MEDIUM
66 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1732
Direction ratios of lines $l_{1}$ and $l_{2}$ are <12,-3, 9> and <4, q,-p> respectively. The values of p and q for which $l_{1}$ and $l_{2}$ are parallel are respectively:
(A)
-1,3
(B)
3,1
(C)
-3,-1
(D)
-1,-3
MEDIUM
67 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1733
The corner points of the feasible region determined by the system of linear constraints are (0,0), (0, 40), (20, 40), (60, 20) and (60, 0). If the objective function of an LPP is $Z=4x+3y$, then the maximum value is :
(A)
200
(B)
300
(C)
240
(D)
120
MEDIUM
68 MCQ_SINGLE 1 Mark(s) 2026
#1734
In the graph, the feasible region representing the Linear Programming Problem for maximising objective function $Z=px+qy$, $p, q>0$ is shaded. If all points on segment AB give max (Z), then which of the following is true? [Graph shows A at (0, 5) and B at (3, 4)]
(A)
$p=2q$
(B)
$p=3q$
(C)
$q=3p$
(D)
$q=2p$
EASY
69 MCQ_SINGLE 1 Mark(s) 2026
#1735
For the feasible region shown below, the non-trivial constraints of the linear programming problem are
(A)
$x+y \le 5$, $x+3y \le 9$
(B)
$x+y \le 5$, $x+3y \ge 9$
(C)
$x+y \ge 5$, $x+3y \le 9$
(D)
$x+y \ge 5$, $3x+y \le 9$
EASY
70 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1736
In a linear programming problem, the linear function which has to be maximized or minimized is called
(A)
a feasible function
(B)
an objective function
(C)
an optimal function
(D)
a constraint
EASY
71 REMEMBER MCQ_SINGLE 1 Mark(s) 2026
#1737
The region represented by the system of inequations $3x+y\ge 3$, $2x-y\ge -5$, $x, y\ge 0$ is:
(A)
unbounded in $1^{st}$ quadrant
(B)
bounded in $1^{st}$ quadrant
(C)
unbounded in $2^{nd}$ quadrant
(D)
bounded in $2^{nd}$ quadrant
EASY
72 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1738
If E and F are two independent events such that $P(E)=\frac{3}{10}$, $P(E\cup F)=\frac{1}{2}$ then $P(E|F)-P(F|E)$ is equal to:
(A)
$\frac{2}{7}$
(B)
$\frac{3}{35}$
(C)
$\frac{1}{70}$
(D)
$\frac{1}{7}$
MEDIUM
73 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1739
For two events A and B such that $P(A) \ne 0$ and $P(B) \ne 1$, $P(A^{\prime}/B^{\prime})=$
(A)
$1-P(A/B)$
(B)
$1-P(A^{\prime}/B)$
(C)
$\frac{1-P(A\cap B)}{P(B^{\prime})}$
(D)
$\frac{1-P(A\cup B)}{P(B^{\prime})}$
MEDIUM
74 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1740
If $3P(A)=P(B)=\frac{3}{5}$ and $P(A|B)=\frac{2}{3}$ then $P(A\cup B)$ is:
(A)
$\frac{3}{5}$
(B)
$\frac{1}{5}$
(C)
$\frac{2}{5}$
(D)
$\frac{2}{15}$
MEDIUM
75 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1741
A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red?
(A)
$\frac{27}{125}$
(B)
$\frac{8}{125}$
(C)
$\frac{2}{125}$
(D)
$\frac{98}{125}$
MEDIUM
76 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1839
The degree of an objective function of a linear programming problem is
(A)
0
(B)
1
(C)
2
(D)
Any natural number
EASY
77 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1840
The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below :
The maximum value of Z - minimum value of Z is
(A)
8
(B)
29
(C)
35
(D)
43
MEDIUM
78 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1841
If $\Delta_{1}=\begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$ and $\Delta_{2}=\begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix}$, then
(A)
$\Delta_{1}=2 \Delta_{2}$
(B)
$\Delta_{2}=-2 \Delta_{1}$
(C)
$\Delta_{1}=\Delta_{2}$
(D)
$\Delta_{2}=-\Delta_{1}$
MEDIUM
79 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1842
One of the values of $x$ for which $\begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1$ is
(A)
$0$
(B)
$\frac{\pi}{4}$
(C)
$\frac{\pi}{3}$
(D)
$\frac{\pi}{2}$
MEDIUM
80 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1843
If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?
(A)
$AB$
(B)
$AB + BA$
(C)
$(A + B)^2$
(D)
$A - B$
MEDIUM
81 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1844
The least value of $f(x) = x^3 - 12x$, $x \in [0, 3]$ is
(A)
$ -16$
(B)
$ -9$
(C)
$ 0$
(D)
$ 16$
MEDIUM
82 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1845
If $\int \frac{3ax}{b^2 + c^2x^2} dx = A \log |b^2 + c^2x^2| + K$, then the value of $A$ is
(A)
$3a$
(B)
$\frac{3a}{2b^2}$
(C)
$\frac{3a}{b^2c^2}$
(D)
$\frac{3a}{2c^2}$
MEDIUM
83 COMPETENCY EVALUATE MCQ_SINGLE 1 Mark(s) 2026
#1846
The value of $\int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1} dx$ is
(A)
0
(B)
log 2
(C)
2 log 2
(D)
$\frac{1}{2}$ log 2
MEDIUM
84 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1847
The area bounded by the curve $y = x|x|$, x-axis and the ordinates $x = -1$ and $x = 1$ is given by
(A)
(B)
$1/3$
(C)
$2/3$
(D)
$4/3$
MEDIUM
85 APPLY MCQ_SINGLE 1 Mark(s) 2026
#1848
The integrating factor of differential equation $R\frac{dx}{dy} + Px = Q$ where P, Q, R are functions of y is
(A)
$e^{\int\frac{P}{Q}dy}$
(B)
$e^{\int Pdy}$
(C)
$e^{\int\frac{P}{R}dy}$
(D)
$e^{\int\frac{P}{R}dx}$
MEDIUM
86 COMPETENCY ANALYZE MCQ_SINGLE 1 Mark(s) 2026
#1849
The order and degree of the differential equation $\frac{d}{dx}(e^y) = 0$ respectively are
(A)
0, 1
(B)
1, 1
(C)
2, 1
(D)
1, not defined
MEDIUM
87 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1850
The value of p for which vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} - p\hat{j} + \hat{k}$ are perpendicular to each other is
(A)
$0$
(B)
$1$
(C)
$\frac{5}{2}$
(D)
$-\frac{5}{2}$
EASY
88 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1851
The value of m for which the points with position vectors $-\hat{i} - \hat{j} + 2\hat{k}$, $2\hat{i} + m\hat{j} + 5\hat{k}$ and $3\hat{i} + 11\hat{j} + 6\hat{k}$ are collinear, is
(A)
8
(B)
-8
(C)
2
(D)
$\frac{5}{2}$
MEDIUM
89 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1852
If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 12$, then the value of $|\vec{a} \cdot \vec{b}|$
(A)
$6\sqrt{3}$
(B)
$8\sqrt{3}$
(C)
$12\sqrt{3}$
(D)
$3\sqrt{12}$
MEDIUM
90 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2026
#1853
The length of perpendicular drawn from point $(2, 5, 7)$ on line $\frac{x}{1}=\frac{y}{0}=\frac{z}{0}$ is
(A)
2
(B)
5
(C)
$\sqrt{74}$
(D)
$\sqrt{78}$
MEDIUM

Assertion & Reason

1 COMPETENCY APPLY ASSERTION_REASON 1 Mark(s) 2026
#1837
Assertion: Lines given by $x = py + q, z = ry + s$ and $x = p'y + q', z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$.
Reason: Two lines $\vec{r} = \vec{a_1} + \lambda \vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu \vec{b_2}$ are perpendicular to each other if $\vec{b_1} \cdot \vec{b_2} = 0$.
(A)
Both A and R are true and R is the correct explanation of A.
(B)
Both A and R are true but R is NOT the correct explanation of A.
(C)
A is true but R is false.
(D)
A is false but R is true.
MEDIUM
2 COMPETENCY EVALUATE ASSERTION_REASON 1 Mark(s) 2026
#1838
Assertion: In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is $\frac{2}{3}$.
Reason: For any two events A and B, $P(A|B) = \frac{P(A \cup B)}{P(B)}$
(A)
Both A and R are true and R is the correct explanation of A.
(B)
Both A and R are true but R is NOT the correct explanation of A.
(C)
A is true but R is false.
(D)
A is false but R is true.
HARD

VSA

1 APPLY VSA 2 Mark(s) 2026
#1494
Check whether function \(f(x)\) defined as

\[
f(x)=
\begin{cases}
\dfrac{|x-3|}{2(x-3)}, & x<3, \\[6pt]
\dfrac{x-6}{6}, & x\ge 3
\end{cases}
\]

is continuous at \(x=3\) or not?
MEDIUM
2 APPLY VSA 2 Mark(s) 2026
#1495
If $ \sqrt{3}\,(x^2+y^2)=4xy$, then find \(\dfrac{dy}{dx}\) at $\left(\frac{1}{2},\,\frac{\sqrt{3}}{2}\right).$
EASY
3 APPLY VSA 1 Mark(s) 2026
#1496
A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm3/min. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is 10 mm, if the semi-vertical angle of conical bottle is $\frac{\pi}{6}$
MEDIUM
4 REMEMBER VSA 2 Mark(s) 2026
#1497
Find the vector of magnitude \(14\) in the direction of \(\overrightarrow{QP}\), where \(P=(1,3,2)\) and \(Q=(-1,0,8)\) respectively.
EASY
5 REMEMBER VSA 2 Mark(s) 2026
#1498
Vectors \(\vec{a}=3\hat{i}-2\hat{j}+2\hat{k}\) and \(\vec{b}=\hat{i}+2\hat{k}\) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.
EASY
6 REMEMBER VSA 2 Mark(s) 2026
#1499
$\text{Simplify: }\tan^{-1}\!\left(\frac{\cos 2x-\sin 2x}{\cos 2x+\sin 2x}\right),\quad 0
EASY
7 REMEMBER VSA 2 Mark(s) 2026
#1500
$\text{Evaluate: }\tan\!\left(\sin^{-1}1-\cos^{-1}\!\left(-\frac{1}{2}\right)\right).$
EASY
8 COMPETENCY APPLY VSA 2 Mark(s) 2026
#1742
Check whether $f:R-\{3\} \rightarrow R$ defined as $f(x)=\frac{x-2}{x-3}$ is onto or not.
MEDIUM
9 VSA 2 Mark(s) 2026
#1743
Check whether $f:Z\times Z \rightarrow Z\times Z$ (where Z is the set of integers) defined as $f(x,y)=(2y,3x)$ is injective or not.
EASY
10 VSA 2 Mark(s) 2026
#1744
A relation R on $A=\{1,2,3\}$ is defined as $R=\{(1,1),(3,3),(1,2)\}$. Is R a symmetric relation? Justify. Write the smallest relation set $R_{1}$ such that $R\cup R_{1}$ becomes an equivalence relation on the set {1, 2, 3}.
EASY
11 VSA 2 Mark(s) 2026
#1745
Find the value of $\sin[\cot^{-1}\sqrt{2}(\cos(\tan^{-1}1))]$.
EASY
12 VSA 2 Mark(s) 2026
#1746
Evaluate : $\tan^{-1}(-\frac{1}{\sqrt{3}})+\cot^{-1}(\frac{1}{\sqrt{3}})+\tan^{-1}(\sin(-\frac{\pi}{2}))+\tan^{-1}(\tan\frac{2\pi}{3})$
EASY
13 VSA 2 Mark(s) 2026
#1747
Evaluate $\sin[\tan^{-1}\tan(\frac{3\pi}{4})]$.
EASY
14 VSA 2 Mark(s) 2026
#1748
Show that the function $f(x)=\begin{cases}\frac{\cos x}{-x+\frac{\pi}{2}}, & x\ne\frac{\pi}{2} \\ 1, & x=\frac{\pi}{2}\end{cases}$ is continuous at $x=\frac{\pi}{2}$.
EASY
15 VSA 2 Mark(s) 2026
#1749
Find whether the function $f(x)=\begin{cases}x-1, & x<2 \\ 2x-3, & x\ge 2\end{cases}$ at $x=2$ is differentiable or not.
EASY
16 VSA 2 Mark(s) 2026
#1750
If $x=a\sin^{3}t$, $y=b\cos^{3}t$, then find $\frac{dy}{dx}$ at $t=\frac{\pi}{4}$.
EASY
17 VSA 2 Mark(s) 2026
#1751
If $x=t+\frac{1}{t}$ and $y=t-\frac{1}{t}$, find $\frac{dy}{dx}$ at $t=2$.
EASY
18 VSA 2 Mark(s) 2026
#1752
Differentiate $x^{x}$ with respect to $x \log x$.
EASY
19 VSA 2 Mark(s) 2026
#1753
If $y=P \cos ux+Q \sin ux$, show that $\frac{d^{2}y}{dx^{2}}+u^{2}y=0$.
EASY
20 VSA 2 Mark(s) 2026
#1754
Find the absolute maximum value of $f(x)=\cos x+\sin^{2}x$, $x \in [0,\pi]$.
EASY
21 VSA 2 Mark(s) 2026
#1755
If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.
EASY
22 VSA 2 Mark(s) 2026
#1756
Find the sub-interval(s) of $(0,\frac{\pi}{2})$ in which $f(x)=\tan x-4x$ is increasing.
EASY
23 VSA 2 Mark(s) 2026
#1757
Find the values of x for which $f(x)=x^{x}$, $x>0$ is increasing.
EASY
24 VSA 2 Mark(s) 2026
#1758
Determine the values of x for which $f(x)=\frac{x-3}{x+1}$, $x\ne -1$ is an increasing function.
EASY
25 VSA 2 Mark(s) 2026
#1759
If $\vec{AB}=\hat{j}+\hat{k}$ and $\vec{AC}=3\hat{i}-\hat{j}+4\hat{k}$ represent the two vectors along the sides AB and AC of $\Delta ABC$, prove that the median $\vec{AD}=\frac{\vec{AB}+\vec{AC}}{2}$ where D is midpoint of BC. Hence, find the length of median AD.
EASY
26 VSA 2 Mark(s) 2026
#1760
If for two unit vectors $\vec{a}$ and $\vec{b}$, $|\vec{a}+2\vec{b}|=|2\vec{a}-\vec{b}|$, then find the angle between $\vec{a}$ and $\vec{b}$.
EASY
27 VSA 2 Mark(s) 2026
#1761
If the position vectors of three points A, B and C are $3\hat{i}+\hat{j}$, $5\hat{i}+6\hat{j}-3\hat{k}$ and $4\hat{j}$ respectively, then show that they form an isosceles triangle.
EASY
28 VSA 2 Mark(s) 2026
#1762
Let two rods placed on the ground be represented by vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.
EASY
29 VSA 2 Mark(s) 2026
#1763
A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with x-axis, $\frac{\pi}{3}$ with y-axis and an acute angle $\theta$ with z-axis. Find $\theta$ and the components of $\vec{a}$.
EASY
30 VSA 2 Mark(s) 2026
#1764
Three honey bees were found flying along the vectors $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}=4\hat{j}-2\hat{k}$ and $\vec{c}=3\hat{i}+2\hat{k}$ respectively. Find the value of $\lambda$ such that the path for $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c}$.
EASY
31 VSA 2 Mark(s) 2026
#1765
If A, B and C be three non-collinear points such that $\vec{AB}=\hat{i}+2\hat{j}-\hat{k}$ and $\vec{AC}=2\hat{i}-3\hat{j}$, then find the area of $\Delta ABC$.
EASY
32 VSA 2 Mark(s) 2026
#1766
Find the co-ordinates of the point on the line $\vec{r}=-\hat{j}+3\hat{k}+\lambda(2\hat{i}-2\hat{j}+\hat{k})$ such that the sum of co-ordinates is 3.
EASY
33 VSA 2 Mark(s) 2026
#1767
If the lines $\frac{x-3}{1}=\frac{1-y}{1}=\frac{z+2}{p}$ and $\frac{2-x}{3}=\frac{y+1}{5}=\frac{z+56}{2p}$ are perpendicular to each other, then find the value(s) of p.
EASY
34 VSA 2 Mark(s) 2026
#1768
Find the vector equation of a line passing through the origin and perpendicular to both the lines $\vec{r}=2\hat{i}-\hat{j}+2\hat{k}+\lambda(3\hat{i}+4\hat{j}+2\hat{k})$ and $\vec{r}=\mu(\hat{i}-\hat{j}+\hat{k})$.
EASY
35 VSA 2 Mark(s) 2026
#1769
Find the angle between the following pair of lines: $\frac{x-2}{3}=\frac{y+5}{2}=\frac{1-z}{-6}$ and $\frac{x-7}{1}=\frac{y}{2}=\frac{6-z}{-2}$.
EASY

SA

1 REMEMBER SA 3 Mark(s) 2026
#1501
$\text{Evaluate: }\int_{0}^{1} x\,\tan^{-1}x\,dx.$
EASY
2 REMEMBER SA 3 Mark(s) 2026
#1502
$\text{Find }\int \sqrt{\frac{x+2}{x-2}}\,dx.$
EASY
3 REMEMBER SA 3 Mark(s) 2026
#1503
$\text{Find: }\int \frac{x^{2}}{(x^{2}+9)(x^{2}+16)}\,dx.$
EASY
4 REMEMBER SA 3 Mark(s) 2026
#1504
If $I_{1}=\int_{-\pi/4}^{\pi/4}\frac{dx}{1+\cos 2x}$ and $I_{2}=\int_{-1/2}^{1/2}|x|\,dx $, then show that $I_{1}-4I_{2}=0$.
EASY
5 COMPETENCY APPLY SA 3 Mark(s) 2026
#1505
Find the general solution of the following differential equation: $x^{2}\frac{dy}{dx}=x^{2}+xy+y^{2}$.
MEDIUM
6 COMPETENCY APPLY SA 3 Mark(s) 2026
#1506
Find the particular solution of the differential equation $x\frac{dy}{dx}=(x+2)(y+2)$, given that $y(1)=-1$.
MEDIUM
7 COMPETENCY APPLY SA 3 Mark(s) 2026
#1508
Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.
MEDIUM
8 COMPETENCY APPLY SA 3 Mark(s) 2026
#1509
The probability of simultaneous occurrence of at least one of the two events $X$ and$Y$ is $a$. If the probability that exactly one of the events $X, Y$ occurs is $b$, prove that $P(X') + P(Y') = 2 – 2a + b$.
MEDIUM
9 COMPETENCY APPLY SA 3 Mark(s) 2026
#1770
Let $A=\mathbb{R}-\{3\}$ and $B=\mathbb{R}-\{1\}$. A function $f:A\rightarrow B$ is defined by $f(x)=(\frac{x-2}{x-3})$. Find whether f is one-one and onto.
MEDIUM
10 COMPETENCY APPLY SA 3 Mark(s) 2026
#1771
Let n be a fixed positive integer. A relation R is defined in set Z such that $R=\{(x,y):(x-y) \text{ is divisible by } n, x, y\in Z\}$. Determine if R is an equivalence relation.
MEDIUM
11 SA 3 Mark(s) 2026
#1772
If $A=\begin{bmatrix}3&2&0\\1&4&0\\0&0&5\end{bmatrix}$, then compute $A^{2}-7A+10~I$.
EASY
12 SA 3 Mark(s) 2026
#1773
If $(\sin x)^{y}=y^{\cos x}$ then find $\frac{dy}{dx}$.
EASY
13 SA 3 Mark(s) 2026
#1774
If $xy=e^{x-y}$, then find $\frac{dy}{dx}$.
EASY
14 SA 3 Mark(s) 2026
#1775
Differentiate $\tan^{-1}(\frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}})$ with respect to $\cos^{-1}x^{2}$.
EASY
15 SA 3 Mark(s) 2026
#1776
A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.
EASY
16 SA 3 Mark(s) 2026
#1777
Find: $\int\frac{x+2}{\sqrt{9x-x^{2}}}dx$
EASY
17 SA 3 Mark(s) 2026
#1778
If $\frac{d}{dx}(F(x))=\frac{1}{e^{x}+1}$ then find $F(x)$ given that $F(0)=\log\frac{1}{2}$
EASY
18 SA 3 Mark(s) 2026
#1779
Find: $\int\frac{dx}{x^{1/2}+x^{1/3}}$
EASY
19 SA 3 Mark(s) 2026
#1780
Find: $\int\tan^{-1}(\frac{1-x}{1+x})dx$
EASY
20 SA 3 Mark(s) 2026
#1781
Find: $\int\frac{\cos x}{(2+\sin x)(4+\sin x)}dx$
EASY
21 SA 3 Mark(s) 2026
#1782
Find: $\int\frac{x+3}{x^{2}+4x+5}dx$
EASY
22 SA 3 Mark(s) 2026
#1783
Find: $\int\frac{x-\sin x}{1-\cos x}dx$
EASY
23 SA 3 Mark(s) 2026
#1784
Evaluate: $\int_{\frac{\pi}{12}}\frac{dx}{1+\sqrt{\cot x}}$
EASY
24 SA 3 Mark(s) 2026
#1785
Evaluate: $\int_{-\frac{\pi}{6}}^{\frac{\pi}{2}}(\sin|x|+\cos|x|)dx$
EASY
25 SA 3 Mark(s) 2026
#1786
Evaluate: $\int_{0}^{\pi}\frac{\sin^{2026}x}{\sin^{2026}x+\cos^{2026}x}dx$
EASY
26 SA 3 Mark(s) 2026
#1787
Evaluate: $\int_{0}^{2}\frac{1}{\sqrt{x^{2}+2x+3}}dx$
EASY
27 SA 3 Mark(s) 2026
#1788
Solve the following differential equation: $x\frac{dy}{dx}=y-x\sin^{2}(\frac{y}{x})$ given that $y(1)=\frac{\pi}{6}$
EASY
28 SA 3 Mark(s) 2026
#1789
Find the general solution of the differential equation: $y\log y\frac{dx}{dy}+x=\frac{2}{y}$
EASY
29 SA 3 Mark(s) 2026
#1790
Find the general solution of the differential equation $2x^{2}\frac{dy}{dx}=y^{2}+2xy.$
EASY
30 SA 3 Mark(s) 2026
#1791
Find a particular solution of the differential equation $(x+1)\frac{dy}{dx}=2 e^{-y}-1$ given that $y=0$ when $x=0$.
EASY
31 SA 3 Mark(s) 2026
#1792
Solve the differential equation $(x+2y^{3})dy=y~dx$.
EASY
32 SA 3 Mark(s) 2026
#1793
Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $5\hat{i}-2\hat{j}$, $5\hat{i}+8\hat{j}$ and $a\hat{i}-52\hat{j}$ respectively, find the value of 'a'.
EASY
33 SA 3 Mark(s) 2026
#1794
If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors, then prove that $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}\le 9$.
EASY
34 SA 3 Mark(s) 2026
#1795
Find a point on the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z-3}{2}$ at a distance of $\sqrt{2}$ units from the point (1, 2, 3).
EASY
35 SA 3 Mark(s) 2026
#1796
Find the shortest distance between the lines $\vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k}$ and $\vec{r}=(1+2\mu)\hat{i}+(2-5\mu)\hat{k}+(4\mu-1)\hat{j}$.
EASY
36 SA 3 Mark(s) 2026
#1797
Solve the following linear programming problem graphically: Maximize $Z=10500x+9000y$ Subject to constraints $x+y \le 50$, $2x+y \le 80$, $x, y \ge 0$
EASY
37 SA 3 Mark(s) 2026
#1798
Solve the following Linear Programming Problem graphically: Maximise $Z=200x+120y$ subject to the constraints $x+y\le 300$, $3x+y\le 600$, $x-y\ge -100$, $x,y\ge 0$
EASY
38 SA 3 Mark(s) 2026
#1799
Solve the following Linear Programming Problem graphically: Maximize $Z=\frac{2x}{5}+\frac{3y}{10}$ subject to constraints $2x+y\le 1000$, $x+y\le 800$, $x, y \ge 0$.
EASY
39 COMPETENCY APPLY SA 3 Mark(s) 2026
#1800
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.
MEDIUM
40 COMPETENCY APPLY SA 3 Mark(s) 2026
#1801
Mother, Father and Son line up at random for a family picture. Let events E: Son on one end and F: Father in the middle. Find $P(E/F)$.
MEDIUM
41 COMPETENCY APPLY SA 3 Mark(s) 2026
#1802
A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that: (i) exactly one surgery is successful, (ii) at most two surgeries are successful.
MEDIUM
42 COMPETENCY APPLY SA 3 Mark(s) 2026
#1803
In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.
MEDIUM
43 COMPETENCY APPLY SA 3 Mark(s) 2026
#1804
A die is rolled. Consider events: $A=\{1,2,5\}$, $B=\{3,5\}$, $C=\{2,3,4,5\}$ and hence find: (i) $P(A|C)$ and $P(C|A)$ (ii) $P(A\cap B|C)$ and $P(A\cup B|C)$.
MEDIUM
44 COMPETENCY APPLY SA 3 Mark(s) 2026
#1805
A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.
MEDIUM
45 APPLY SA 3 Mark(s) 2026
#1836
Solve the following linear programming problem graphically :
Minimize $Z = 13x - 15y$
Subject to constraints
$$x + y \leq 7,$$
$$2x - 3y + 6 \geq 0,$$
$$x \geq 0, y \geq 0$$
MEDIUM

LA

1 COMPETENCY APPLY LA 5 Mark(s) 2026
#1806
Show that $f:R\rightarrow R$ defined as $f(x)=\frac{x}{\sqrt{1+x^{2}}}$ is one-one but not onto.
MEDIUM
2 COMPETENCY APPLY LA 5 Mark(s) 2026
#1807
Find the domain of $g(x)=\cos^{-1}(x^{2}-1)$. Hence, find the value of x for which $g(x)=\frac{\pi}{3}$. Also, write the range of $\cos^{-1}x$ other than its principal branch.
MEDIUM
3 COMPETENCY APPLY LA 5 Mark(s) 2026
#1808
If $P=\begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}$ and $Q=\begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix}$, find (QP) and hence solve the following system of equations using matrices: $x-y=3$, $2x+3y+4z=17$, $y+2z=7$
MEDIUM
4 COMPETENCY APPLY LA 5 Mark(s) 2026
#1809
Obtain the value of $\Delta=\begin{vmatrix}1+x&1&1\\1&1+y&1\\1&1&1+z\end{vmatrix}$ in terms of x, y, and z. Further, if $\Delta=0$ and x, y, z are non-zero real numbers, prove that $x^{-1}+y^{-1}+z^{-1}=-1$
MEDIUM
5 COMPETENCY APPLY LA 5 Mark(s) 2026
#1810
On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.
MEDIUM
6 COMPETENCY APPLY LA 5 Mark(s) 2026
#1811
Given that $P=\begin{bmatrix}2&-1\\3&4\end{bmatrix}$, $Q=\begin{bmatrix}5&2\\7&4\end{bmatrix}$ and $R=\begin{bmatrix}2&5\\3&8\end{bmatrix}$, find a matrix S such that PQ - RS is a null matrix.
MEDIUM
7 COMPETENCY APPLY LA 5 Mark(s) 2026
#1812
A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.
MEDIUM
8 COMPETENCY APPLY LA 5 Mark(s) 2026
#1813
If $y\sqrt{x^{2}+1}=\log\sqrt{x^{2}+1}-x$, show that $(x^{2}+1)\frac{dy}{dx}+xy+1=0$.
MEDIUM
9 COMPETENCY APPLY LA 5 Mark(s) 2026
#1814
Find the differential of $x^{\cot x}+\frac{2x^{2}-3}{2x^{2}-x+2}$ with respect to x.
HARD
10 COMPETENCY APPLY LA 5 Mark(s) 2026
#1815
Find the sub intervals in which $f(x)=\cot^{-1}(\sin x+\cos x)$, $x \in (0,\pi)$ is increasing and decreasing.
HARD
11 COMPETENCY APPLY LA 5 Mark(s) 2026
#1816
A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.
HARD
12 COMPETENCY APPLY LA 5 Mark(s) 2026
#1817
Find: $\int\frac{x}{(x-1)(x^{2}+4)}dx$
MEDIUM
13 COMPETENCY APPLY LA 5 Mark(s) 2026
#1818
Evaluate: $\int_{0}^{1}\frac{x~\tan^{-1}x}{(1+x^{2})^{3/2}}dx$
HARD
14 COMPETENCY APPLY LA 5 Mark(s) 2026
#1819
Using integration, find the area of the region enclosed by the curve $y=|x-6|$, the x-axis, and between $x=4$ and $x=8$.
MEDIUM
15 COMPETENCY APPLY LA 5 Mark(s) 2026
#1820
Sketch the curve $\{(x, y): 100x^{2}+25y^{2}=2500\}$ and find the area of the region enclosed by it, using integration.
MEDIUM
16 COMPETENCY APPLY LA 5 Mark(s) 2026
#1821
Solve the differential equation $y~e^{y}dx = (y^{3} + 2x~e^{y}) dy$, when $y(0)=1$.
HARD
17 COMPETENCY APPLY LA 5 Mark(s) 2026
#1822
Find the general solution of the differential equation $(x^{3}-3xy^{2})dx=(y^{3}-3x^{2}y)dy$.
HARD
18 COMPETENCY APPLY LA 5 Mark(s) 2026
#1823
A line passing through the points $A(1,2,3)$ and $B(5,8,11)$ intersects the line $\vec{r}=4\hat{i}+\hat{j}+\lambda(5\hat{i}+2\hat{j}+\hat{k})$. Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.
HARD
19 COMPETENCY APPLY LA 5 Mark(s) 2026
#1824
Represent the equations of lines $l_{1}$ and $l_{2}$ in vector form and check whether they are intersecting or not. $l_{1}:\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, $l_{2}:\frac{x+1}{-1}=\frac{2-y}{-2}=\frac{z-5}{5}$
MEDIUM
20 COMPETENCY APPLY LA 5 Mark(s) 2026
#1825
Opposite sides of a square are along the lines : $\vec{r}=\hat{i}+2\hat{j}-4\hat{k}+\lambda(2\hat{i}+3\hat{j}+6\hat{k})$ and $\vec{r}=3\hat{i}+3\hat{j}-5\hat{k}+\mu(2\hat{i}+3\hat{j}+6\hat{k})$. Find the area of the square if direction ratios of other pair of opposite sides of the square are given by <-3, 6, p>. Also, find the value of p.
HARD
21 COMPETENCY APPLY LA 5 Mark(s) 2026
#1826
Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ and is parallel to the vector $3\hat{i}+2\hat{j}-8\hat{k}$.
HARD
22 COMPETENCY APPLY LA 5 Mark(s) 2026
#1827
Find the foot of the perpendicular from the point (0, 2, 3) on the line $\frac{-x-3}{-5}=\frac{1-y}{-2}=\frac{3z+12}{9}$ and hence find the length of the perpendicular.
HARD
23 COMPETENCY APPLY LA 5 Mark(s) 2026
#1828
Find the value of p if the shortest distance between the lines $\vec{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$ and $\vec{r}=(p\hat{i}-\hat{j}-\hat{k})+\mu(2\hat{i}+\hat{j}+2\hat{k})$ is $\frac{3}{\sqrt{2}}$ units.
HARD
24 COMPETENCY APPLY LA 5 Mark(s) 2026
#1829
Solve the following Linear Programming Problem graphically: Maximise $Z=600x+400y$ subject to the constraints $x+2y\le12$, $2x+y\le12$, $4x+5y\ge20$, $x, y\ge0$
MEDIUM
25 APPLY LA 5 Mark(s) 2026
#1830
Check whether the lines given by $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-4}{5} = \frac{y-1}{2} = z$ are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.
MEDIUM
26 APPLY LA 5 Mark(s) 2026
#1831
If $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations :
$-2y + z = 7, 2x - y - z = 8, x - 2y = 10$
MEDIUM
27 APPLY LA 5 Mark(s) 2026
#1832
If $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$ is a singular matrix, then find all values of $x$
where $x \in \left[0, \frac{\pi}{2}\right]$.
MEDIUM
28 APPLY LA 5 Mark(s) 2026
#1833
If $x = \cos t, y = \cos mt$, prove that $\left(1-x^2\right) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2y = 0$.
MEDIUM
29 APPLY LA 5 Mark(s) 2026
#1834
A relation R is defined on Z, the set of integers, as $R = \{(x, y) : |x - y|\text{ is divisible by a prime number 'p', } x, y \in \mathbb{Z}\}$
check whether R is an equivalence relation or not.
MEDIUM
30 APPLY LA 5 Mark(s) 2026
#1835
A function $f : \mathbb{R} - \left\{\frac{3}{5}\right\} \longrightarrow \mathbb{R} - \left\{\frac{3}{5}\right\}$ is defined as $f(x) = \frac{3x + 2}{5x - 3}$. Show that f is one-one and onto.
MEDIUM