Sample QuestionsModel Paper 7 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The total number of 9 digit numbers which have all different digits is
- A
- B
$10 \times 10!$
- C
- D
$9 \times 9$ !
View full solution →If $y=\frac{\sin x+\cos x}{\sin x-\cos x}$, then $\frac{d y}{d x}$ at $x =0$ is equal to
Answer: B.
View full solution →$\cos 15^{\circ}=?$
- A
$\frac{(\sqrt{3}+1)}{\sqrt{2}}$
- B
$\frac{(\sqrt{3}+1)}{2 \sqrt{2}}$
- C
$\frac{(\sqrt{3}-1)}{2 \sqrt{2}}$
- D
$\frac{(\sqrt{3}-1)}{\sqrt{2}}$
View full solution →If A = {1, 2, 3, 4, 5, 6} then the number of proper subsets is
Solve the system of inequalities: $\frac{x+7}{x-8}>2, \frac{2 x+1}{7 x-1}>5$
View full solution →Assertion (A): A sequence is said to definite if it has finite no of terms.
Reason (R): The sequence whose $n^{\text {th }}$ term if $\frac{2^n}{n}$ if $2,2, \frac{8}{3}, 4 \ldots$
- 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.
View full solution →Assertion (A): Let $A =\{ a , b \}$ and $B =\{ a , b , c \}$. Then, $A \not \subset B$.
Reason (R): If $A \subset B$, then $A \cup B=B$.
- 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.
View full solution →Find the coordinates of the point which divides the join of $A(-5, 11)$ and $B(4, -7)$ in the ratio $2 : 7.$
View full solution →Is $A =\{ x : x \in N , 1< x \leq 2\}$ null set?
View full solution →If $A$ and $B$ are two events associated with a random experiment such that $P ( A )=0.25, P ( B )=0.4$ and $P ( A$ or $B )= 0.5 ,$ find the values of
$i. P ( A$ and B $)$
$ii. P ( A$ and $\bar{B})$
View full solution →Two dice are thrown simultaneously. Find the probability of getting a total of at least $10.$
View full solution →Evaluate: $\lim _{\theta \rightarrow 0} \frac{1-\cos 4 \theta}{1-\cos 6 \theta}$.
View full solution →Out of $25$ members in a family, $12$ like to take tea, $15$ like to take coffee and $7$ like to take coffee and tea both. How many like
$i.$ at least one of the two drinks
$ii.$ only tea but not coffee
$iii.$ only coffee but not tea
$iv.$ neither tea nor coffee
View full solution →The sum of first three terms of a $G.P.$ is $\frac{39}{10}$ and their product is $1.$ Find the common ratio and the terms.
View full solution →The sum of three numbers in $G.P.$ is $14.$ If the first two terms are each increased by $1$ and the third term decreased by $1,$ the resulting numbers are in $A.P.$ Find the numbers.
View full solution →Differentiate the function: $3^{ x -5}$
View full solution →If f(x) = mx + c and f(0) = f'(0) = 1. What is value of f(2)?
The conjugate of a complex number $z$, is the complex number, obtained by changing the
sign of imaginary part of $z$ . It is denoted by $\bar{z}$.
The modulus $($or absolute value$)$ of a complex number, $z = a +$ ib is defined as the non
negative real number
$\sqrt{a^2+b^2}$. It is denoted by $|z|$. i.e.
$|z|=\sqrt{a^2+b^2}$
Multiplicative inverse of $z$ is $\frac{\bar{z}}{|z|^2}$. It is also called reciprocal of $z .$
$z \bar{z}=|z|^2$
$i.$ If $f(z)=\frac{7-z}{1-z^2}$, where $z=1+2 i$, then find $|f(z)|. ( 1 )$
$ii.$ Find the value of $(z+3)(\bar{z}+3). (1)$
$iii.$ If ( $x-i y)(3+5 i)$ is the conjugate of $-6-24 i$, then find the value of $x+y. (2)$
OR
If $z=3+4 i$, then find $\bar{z}. (2)$
View full solution →Two students Ankit and Vinod appeared in an examination. The probability that Ankit will qualify the examination is $0.05$ and that Vinod will qualify is $0.10.$ The probability that both will qualify is $0.02.$
$i.$ Find the probability that atleast one of them will qualify the exam. $(1)$
$ii.$ Find the probability that atleast one of them will not qualify the exam. $(1)$
$iii.$ Find the probability that both Ankit and Vinod will not qualify the exam. $(2)$
OR
Find the probability that only one of them will qualify the exam. $(2)$
View full solution →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): 2a + b = 5; a, b ∈ W} as the set of ordered pairs (in roster form). (2) View full solution →Prove that: $\cos 36^{\circ} \cos 42^{\circ} \cos 60^{\circ} \cos 78^{\circ}=\frac{1}{16}$.
View full solution →If $\sin x=\frac{\sqrt{5}}{3}$ and $x$ lies in the 2nd quadrant, find the values of $\cos \frac{x}{2}, \sin \frac{x}{2}$ and $\tan \frac{x}{2}$.
View full solution →Solve the following system of linear inequalities. $2(2 x +3)-10<6( x -2)$ and $\frac{2 x-3}{4}+6 \geq 2+\frac{4 x}{3}$
View full solution →Find the lengths major and minor axes, coordinates of the vertices, coordinates of the foci, eccentricity, and length of the latus rectum of the ellipse $9 x^2+y^2=36$
View full solution →Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. $\frac{x^2}{100}+\frac{y^2}{400}=1$
View full solution →