If $P(n, r) = C(n, r)$ then
- A$r = 0$ or $2$
- B$r = 1$ or $ n$
- ✓$r = 0$ or $1$
- D$n = r$
Answer: C.
View full solution →Question types
Model Paper 5 question types
45 questions across 6 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
45
Questions
6
Question groups
5
Question types
01
M.C.Q (1 Marks)
18 Q→02Assertion (A) & Reason (B) MCQ
2 Q→032 Marks Questions
7 Q→043 Marks Question
9 Q→05Case study (4 Marks)
3 Q→065 Marks Questions
6 Q→Sample Questions
Model Paper 5 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $f(x)=\sqrt{1-x^2}, x \in(0,1)$, then $f^{\prime}(x)$, is equal to
- A$\sqrt{1-x^2}$
- B$\sqrt{x^2-1}$
- C$\frac{1}{\sqrt{1-x^2}}$
- D$\frac{-x}{\sqrt{1-x^2}}$
The value of $\sin 78^{\circ}-\sin 66^{\circ}-\sin 42^{\circ}+\sin 60^{\circ}$ is
- ✓None of these
- B$\frac{1}{2}$
- C$-1$
- D$\frac{-1}{2}$
Answer: A.
View full solution →Let R be set of points inside a rectangle of sides a and b (a, b > 1) with two sides along the positive direction of x-axis and y-axis. Then
- A$R=\{(x, y): 0 \leq x \leq a, 0 \leq y \leq b\}$
- B
- C
- D
Solve: $3 x+5< x-7$, when $x$ is a real number
- A$x>-12$
- B$x<-12$
- ✓$x<-6$
- D$x>-6$
Answer: C.
View full solution →Assertion $(A):$ The sum of first $6$ terms of the $GP 4, 16, 64,...$ is equal to $5460.$
Reason $(R):$ Sum of first $n$ terms of the $G.P$ is given by $S _{ n }=\frac{a\left(r^n-1\right)}{r-1}$,
where $a =$ first term $r =$ common ratio and $|r|>1$.
Reason $(R):$ Sum of first $n$ terms of the $G.P$ is given by $S _{ n }=\frac{a\left(r^n-1\right)}{r-1}$,
where $a =$ first term $r =$ common ratio and $|r|>1$.
- ✓Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
- BBoth $A$ 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.
Answer: A.
View full solution →Assertion (A): if A = set of letters in Alloy B = set of letters in LOYAL, then set A & B are equal sets.
Reason (R): If two sets have exactly the same elements, they are called equal sets.
Reason (R): If two sets have exactly the same elements, they are called equal sets.
- ABoth A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A.
- CA is true but R is false.
- DA is false but R is true.
Find the equations of the lines which cut$-$off intercepts on the axes whose sum and product are $1$ and $-6$ respectively.
View full solution →What is represented by the shaded regions in each of the following Venn$-$diagrams.
View full solution →A five digit number is formed by the digits 1, 2, 3, 4, 5 without repetition. Find the probability that the number is divisible by 4.
Determine the probability p, for event. An odd number appears in a single toss of a fair die
Evaluate: $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{x^2}$.
View full solution →For any sets $A$ and $B$ show that
$i. (A \cap B) \cup(A-B)=A$
$ii. A \cup(B-A)=A \cup B$
View full solution →$i. (A \cap B) \cup(A-B)=A$
$ii. A \cup(B-A)=A \cup B$
If the sum of an infinite decreasing $G.P.$ is $3$ and the sum of the squares of its term is $\frac{9}{2}$, then write its first term and common difference.
View full solution →If $p^{th},$ qth and rth terms of an $A.P$. and $G.P$. are both $a, b,$ and $c$ respectively. Show that
$a^{b-c} \cdot b^{c-a} \cdot c^{a-b}=1$
View full solution →$a^{b-c} \cdot b^{c-a} \cdot c^{a-b}=1$
Evaluate $\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}$
View full solution →Evaluate $\lim _{x \rightarrow 2}\left(\frac{x^2-4}{\sqrt{x+2}-\sqrt{3 x-2}}\right)$
View full solution →Read the following text carefully and answer the questions that follow:
A number of the form $Z=x+i y$, where $x$ and $y$ are real and $i=\sqrt{-1}$ is called a complex number. Consider the complex number $Z_1=2+3 i$ and $Z_2=4-3 i$.
$i$. Find the imaginary part of $Z_1 \overline{Z_1}... (1)$
$ii$. Find the real part of $\frac{z_1}{z_2}. (1)$
$iii$. Find the imaginary part of $Z_1-Z_2. (2)$
OR
Find the real part of $Z _1. (2)$
View full solution →A number of the form $Z=x+i y$, where $x$ and $y$ are real and $i=\sqrt{-1}$ is called a complex number. Consider the complex number $Z_1=2+3 i$ and $Z_2=4-3 i$.
$i$. Find the imaginary part of $Z_1 \overline{Z_1}... (1)$
$ii$. Find the real part of $\frac{z_1}{z_2}. (1)$
$iii$. Find the imaginary part of $Z_1-Z_2. (2)$
OR
Find the real part of $Z _1. (2)$
There are $4$ red, $5$ blue and $3$ green marbles in a basket.
$i.$ If two marbles are picked at randomly, find the probability that both red marbles. $(1)$
$ii.$ If three marbles are picked at randomly, find the probability that all green marbles. $(1)$
$iii.$ If two marbles are picked at randomly then find the probability that both are not blue marbles. $(2)$
OR
If three marbles are picked at randomly, then find the probability that atleast one of them is blue. $(2)$
View full solution →$i.$ If two marbles are picked at randomly, find the probability that both red marbles. $(1)$
$ii.$ If three marbles are picked at randomly, find the probability that all green marbles. $(1)$
$iii.$ If two marbles are picked at randomly then find the probability that both are not blue marbles. $(2)$
OR
If three marbles are picked at randomly, then find the probability that atleast one of them is blue. $(2)$
Read the following text carefully and answer the questions that follow: Representation of a Relation
A relation can be represented algebraically by roster form or by set $-$ builder form and visually it can be represented by an arrow diagram which are given below
$i$. Roster form In this form, we represent the relation by the set of all ordered pairs belongs to $R$.
$ii$. Set-builder form In this form, we represent the relation $R$ from set $A$ to set $B$ as $R=\{(a, b): a \in A, b \in B\}$ and the rule which relate the elements of $A$ and $B$.
$iii$. Arrow diagram To represent a relation by an arrow diagram, we draw arrows from first element to second element of all ordered pairs belonging to relation $R$.
Questions
$i$. If $n(A) = 3$ and $B = \{2, 3, 4, 6, 7, 8\}$ then find the number of relations from $A$ to $B. (1)$
$ii$. If $A = {a, b}$ and $B = {2, 3},$ then find the number of relations from $A$ to $B. (1)$
$iii$. If $A = {a, b}$ and $B = {2, 3},$ write the relation in set-builder form. $(2)$
OR
Express of $R =\{( a , b ): 2 a + b =5 ; a , b \in W \}$ as the set of ordered pairs $($in roster form$). (2)$
View full solution →A relation can be represented algebraically by roster form or by set $-$ builder form and visually it can be represented by an arrow diagram which are given below
$i$. Roster form In this form, we represent the relation by the set of all ordered pairs belongs to $R$.
$ii$. Set-builder form In this form, we represent the relation $R$ from set $A$ to set $B$ as $R=\{(a, b): a \in A, b \in B\}$ and the rule which relate the elements of $A$ and $B$.
$iii$. Arrow diagram To represent a relation by an arrow diagram, we draw arrows from first element to second element of all ordered pairs belonging to relation $R$.
Questions
$i$. If $n(A) = 3$ and $B = \{2, 3, 4, 6, 7, 8\}$ then find the number of relations from $A$ to $B. (1)$
$ii$. If $A = {a, b}$ and $B = {2, 3},$ then find the number of relations from $A$ to $B. (1)$
$iii$. If $A = {a, b}$ and $B = {2, 3},$ write the relation in set-builder form. $(2)$
OR
Express of $R =\{( a , b ): 2 a + b =5 ; a , b \in W \}$ as the set of ordered pairs $($in roster form$). (2)$
Prove that: $\sin 5 x=5 \sin x-20 \sin ^3 x+16 \sin ^5 x$.
View full solution →Prove that $\cos 12^{\circ}+\cos 60^{\circ}+\cos 84^{\circ}=\cos 24^{\circ}+\cos 48^{\circ}$
View full solution →Solve for $x , \frac{|x+3|+x}{x+2} > 1 zZ$
View full solution →Show that the equation $x^2-2 y^2-2 x+8 y-1=0$ represents a hyperbola. Find the coordinates of the centre, lengths of the axes, eccentricity, latusrectum, coordinates of foci and vertices and equations of directrices of the hyperbola.
View full solution →Find the $(i)$ lengths of major and minor axes, $(ii)$ coordinate of the vertice, $(iii)$ coordinate of the foci, $(iv)$ eccentricity, and $(v)$ length of the latus rectum of ellipe: $16 x^2+25 y^2=400$.
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