The value of $\left({ }^7 C_0+{ }^7 C_1\right)+\left({ }^7 C_1+{ }^7 C_2\right)+\ldots .+\left({ }^7 C_6+{ }^7 C_7\right)$ is
- ✓$2^8-2$
- B$2^8=1$
- C$2^7-1$
- D$2^8$
Answer: A.
View full solution →Question types
Model Paper 8 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 8 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
$\lim _{x \rightarrow \pi} \frac{\sin x}{x-\pi}$ is equal to
- A1
- B-1
- C2
- D-2
If $3 \sin x + 4 \cos x = 5,$ then $4 \sin x - 3 \cos x =$
- A$1$
- B$5$
- C$3$
- ✓$0$
Answer: D.
View full solution →Which of the following is a set?
A. A collection of vowels in English alphabets is a set.
B. The collection of most talented writers of India is a set.
C. The collection of most difficult topics in Mathematics is a set.
D. The collection of good cricket players of India is a set.
A. A collection of vowels in English alphabets is a set.
B. The collection of most talented writers of India is a set.
C. The collection of most difficult topics in Mathematics is a set.
D. The collection of good cricket players of India is a set.
- AB
- BD
- CA
- DC
If $x$ is a real number and $|x|<3$, then
- A- 3 < x < 3
- B$x \geq-3$
- C$x \geq 3$
- D$-3 \leq x \leq 3$
Assertion $(A):$ If the sum of first two terms of an infinite $GP$ is $5$ and each term is three times the sum of the succeeding terms, then the common ratio is $\frac{1}{4}$.
Reason $(R):$ In an $AP\ 3, 6, 9, 12 .........$ the $10^{th}$ term is equal to $33.$
Reason $(R):$ In an $AP\ 3, 6, 9, 12 .........$ the $10^{th}$ term is equal to $33.$
- 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 $A.$
- ✓$A$ is true but $R$ is false
- D$A$ is false but $R$ is true.
Answer: C.
View full solution →Assertion (A): The collection of all natural numbers less than 100' is a set.
Reason (R): A set is a well-defined collection of the distinct objects.
Reason (R): A set is a well-defined collection of the distinct objects.
- 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 length of perpendicular from the origin to the lines $7x + 24y = 50.$
View full solution →In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?
Check whether the probabilities $P ( A )$ and $P ( B )$ are consistently defined $P ( A )=0.5, P ( B )=0.4, P(A \cup B)=0.8$
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 →$i. P ( A$ and $B )$
$ii. P ( A$ and $\bar{B})$
Evaluate $\lim _{x \rightarrow 1} \frac{x^3-1}{x-1}$
View full solution →Are the $E=\left\{x: x \in Z, x^2 \leq 4\right\}$ and $F=\left\{x: x \in Z, x^2=4\right\}$ pairs of equal set?
View full solution →If $A.M$. and $G.M$. of roots of a quadratic equation are $8$ and $5$ respectively then obtain the quadratic equation.
View full solution →he sum of three numbers $a, b, c$ in $A.P.$ is $18.$ If a and b are each increased by $4$ and $c$ is increased by $36,$ the new numbers form a $G.P.$ Find $a, b, c.$
View full solution →Differentiate $e ^{ ax + b }$ from first principle.
View full solution →Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{2 x}{\sqrt{a+x}-\sqrt{a-x}}$.
View full solution →Two complex numbers $Z _1= a + ib$ and $Z _2= c + id$ are said to be equal, if $a = c$ and $b = d$.
$i.$ If $(x+i y)(2-3 i)=4+i$ then find the value of $(x, y)$.
$ii.$ If $\frac{(1+i)^2}{2-i}=x+i y$, then find the value of $x+y. (1)$
$iii.$ If $\left(\frac{1-i}{1+i}\right)^{100}=a+i b$, then find the values of $a$ and $b. (2)$
$OR$
If $(a-2,2 b+1)=(b-1, a+2)$, then find the real values of $a$ and $b. (2)$
View full solution →$i.$ If $(x+i y)(2-3 i)=4+i$ then find the value of $(x, y)$.
$ii.$ If $\frac{(1+i)^2}{2-i}=x+i y$, then find the value of $x+y. (1)$
$iii.$ If $\left(\frac{1-i}{1+i}\right)^{100}=a+i b$, then find the values of $a$ and $b. (2)$
$OR$
If $(a-2,2 b+1)=(b-1, a+2)$, then find the real values of $a$ and $b. (2)$
On her vacation, Priyanka visits four cities. Delhi, Lucknow, Agra, Meerut in a random order.
$i$. What is the probability that she visits Delhi before Lucknow? $(1)$
$ii$. What is the probability she visit Delhi before Lucknow and Lucknow before Agra? $(1)$
$iii.$ What is the probability she visits Delhi first and Lucknow last? $(2)$
OR
What is the probability she visits Delhi either first or second? $(2)$
View full solution →$i$. What is the probability that she visits Delhi before Lucknow? $(1)$
$ii$. What is the probability she visit Delhi before Lucknow and Lucknow before Agra? $(1)$
$iii.$ What is the probability she visits Delhi first and Lucknow last? $(2)$
OR
What is the probability she visits Delhi either first or second? $(2)$
Method to Find the Sets When Cartesian Product is Given
For finding these two sets, we write first element of each ordered pair in first set say $A$ and corresponding second element in second set $B\ ($say$)$.
Number of Elements in Cartesian Product of Two Sets
If there are $p$ elements in set $A$ and $q$ elements in set $B$, then there will be pq elements in $A \times B$ i.e. if $n(A)=p$ and $n(B)=q$, then $n(A \times B)=p q$.
$i$. The Cartesian product $A \times A$ has $9$ elements among which are found $(-1,0)$ and $(0,1)$. Find the set $A$ and the remaining elements of $A \times A. (1)$
$ii. A$ and $B$ are two sets given in such a way that $A \times B$ contains $6$ elements. If three elements of $A \times B$ are $(1, 3 ), (2,5)$ and $(3, 3),$ then find the remaining elements of $A \times B. (1)$
$iii$. If the set $A$ has $3$ elements and set $B$ has $4$ elements, then find the number of elements in $A \times B$. $(2)$
OR
If $A \times B=\{(a, 1),(b, 3),(a, 3),(b, 1),(a, 2),(b, 2)\}$. Find $A$ and $B .(2)$
View full solution →For finding these two sets, we write first element of each ordered pair in first set say $A$ and corresponding second element in second set $B\ ($say$)$.
Number of Elements in Cartesian Product of Two Sets
If there are $p$ elements in set $A$ and $q$ elements in set $B$, then there will be pq elements in $A \times B$ i.e. if $n(A)=p$ and $n(B)=q$, then $n(A \times B)=p q$.
$i$. The Cartesian product $A \times A$ has $9$ elements among which are found $(-1,0)$ and $(0,1)$. Find the set $A$ and the remaining elements of $A \times A. (1)$
$ii. A$ and $B$ are two sets given in such a way that $A \times B$ contains $6$ elements. If three elements of $A \times B$ are $(1, 3 ), (2,5)$ and $(3, 3),$ then find the remaining elements of $A \times B. (1)$
$iii$. If the set $A$ has $3$ elements and set $B$ has $4$ elements, then find the number of elements in $A \times B$. $(2)$
OR
If $A \times B=\{(a, 1),(b, 3),(a, 3),(b, 1),(a, 2),(b, 2)\}$. Find $A$ and $B .(2)$
If $A + B + C =\pi,$ prove that $\frac{\sin 2 A+\sin 2 B+\sin 2 C}{\sin A+\sin B+\sin C}=8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$
View full solution →Prove that: $\cot x+\cot \left(\frac{\pi}{3}+x\right)+\cot \left(\frac{2 \pi}{3}+x\right)=3 \cot 3 x$
View full solution →Solve the following system of linear inequalities $\frac{4 x}{3}-\frac{9}{4}x$
View full solution →A visitor with sign board 'DO NOT LITTER' is moving on a circular path in an exhibition. During the movement he stops at points represented by (3, - 2) and (-2, 0). Also, centre of the circular path is on the line 2x - y = 3. What is the equation of the path? What message he wants to give to the public?
Fine the lengths major and minor axes, coordinates of the vertices, coordinates of the foci, eccentricity, and length of the latus rectum of the ellipse $25 x^2+4 y^2=100$.
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