TS EAPCET (Engineering) Binomial Theorem — practice questions
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Practice TS EAPCET (Engineering) Binomial Theorem in the app →A fair coin is tossed twice. How many outcomes are there in the sample space?A fair six-sided die is rolled once. What is the probability of getting an even number?$ in(180° - \theta)$ equals:$ in^2\theta + \cos^2\theta$ equals:$ in(A + B) = $$\cos 2\theta$ equals:If $ in\theta + \cos\theta = 1$, then $ in\theta \cdot \cos\theta$ equals:$\cos 3\theta$ equals:A **random variable** is best defined as:A **discrete** random variable is one whose possible values:For a continuous random variable $X$, the probability that $X$ equals any specific value is:The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:When two fair dice are rolled and $X$ denotes the sum of the upper faces, the range of $X$ is:If a fair coin is tossed twice and $X$ = number of heads, then $P[X = 1]$ equals:If two cards are drawn with replacement from a deck of 52, $P[X = 2]$ where $X$ = number of aces equals:A fair die is thrown. Let $X$ = number of factors of the number on the upper face. $P[X = 2]$ equals:$f(x) = 3x^2$ for $0 < x < 1$ is the p.d.f. of $X$. The probability $P[1/2 < X < 1]$ is:If $f(x) = kx^2(1 - x)$ for $0 < x < 1$ is the p.d.f. of $X$, the value of $k$ is:For p.d.f. $f(x) = x/8$, $0 < x < 4$, the probability $P[x < 1.5]$ is:Find $k$ if $f(x) = kx$ for $0 < x < 2$ and 0 otherwise is a valid p.d.f.The c.d.f. $F(x)$ of the p.d.f. $f(x) = 3x^2$ for $0 < x < 1$ is:Two cards are drawn with replacement. $X$ = number of aces. The probability $P[X \geq 1]$ equals:For p.d.f. $f(x) = x^2/3$ for $-1 < x < 2$, the probability $P[0 < X \leq 1]$ equals:For p.d.f. $f(x) = k(4 - x^2)$ for $-2 \leq x \leq 2$ (and 0 elsewhere), the value of $k$ is:For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:If a fair coin is tossed $n$ times, the probability of any exact pattern (like all heads) is:If a die is thrown and 'getting an odd number' is success, then $p$ and $q$ are:A bag has balls marked 0–9. Four balls drawn with replacement. Probability that **none** is marked '0' is:Probability of bomb hitting target = 0.8. Out of 10 bombs dropped, probability that **exactly 2 miss** equals sin(0°) equals:cos(90°) equals:sin²θ + cos²θ equals:tan(45°) equals:In which quadrant is sin θ positive and cos θ negative?sec θ is defined as:sin(A + B) equals:cos(A + B) equals:sin(60°) cos(30°) + cos(60°) sin(30°) equals:If sin θ = 3/5 (θ in Q1), find cos θ:Find cos(75°) using cos(A + B):sin(2θ) equals:Find sin(15°):If tan θ = 1/2 (acute), find sin 2θ:For 0 ≤ θ < 2π, solutions of sin θ = 1/2:1 - cos²θ equals:sin(A) + sin(B) equals (sum-to-product):In a right triangle, if hypotenuse = 13 and one leg = 5, sin θ for angle θ opposite to that leg:tan(A + B) equals:sin(180° - θ) equals:In triangle ABC with sides a, b, c opposite to angles A, B, C: sine rule states:If sin θ + cos θ = 1, then sin θ × cos θ equals:General solution of cos θ = 0:If tan θ + cot θ = 2, find sin 2θ:If sin θ = 12/13 (acute θ in Q1), find tan(θ/2):Minimum value of sin²θ + cos⁴θ:If sin θ = 12/13 (acute), find tan(θ/2):In triangle with sides 7, 8, 9, find cos of the angle opposite to side 9 (use cosine rule):General solution of sin θ = sin α:The arithmetic mean of n values x1, x2, ..., xn is:Median of an odd-length sorted data set is:Probability of a certain event:Probability of rolling a 6 on a fair die:For independent events A and B, P(A and B) equals:A binomial raised to the ninth power expands into how many terms?How many terms does the expansion of (2x + 5)^12 contain?In the expansion of (a + b)^n, the power of b in the first term isEvery term in the expansion of (x + y)^7 has a total degree ofIn (a + b)^3, the term containing a squared times b carries the numberAmong the entries 1, 4, 6, 4, 1 for index 4, the centre one isThe mathematician behind Pascal's triangle belonged to which country?In ancient Indian mathematics, Pascal's triangle was known asWhich ancient Indian scholar gave the Meru-Prastara arrangement of coefficients?Adding every number in the fifth-index row of Pascal's triangle givesWhich number multiplies x^5 once (2x + 3y)^5 is fully expanded?After expanding (2x + 3y)^5, the x^4 y term is multiplied byExpand (2x + 3y)^5; the number attached to x^3 y^2 isWhich number sits in front of y^5 when (2x + 3y)^5 is written out?Expanding (x + 2)^6 gives a cubic term whose multiplier isSetting x = 0 in (x + 2)^6 leaves the valueThe linear term of (x + 2)^6 has multiplierPutting x = 1 in (x + 2)^6, the sum of all its coefficients isFor (x - 2y)^5, find the multiplier of the x^3 y^2 termIn the expansion of (x - y)^n, the signs of successive termsThe value of 7C0 + 7C1 + 7C2 + ... + 7C7 isThe alternating sum nC0 - nC1 + nC2 - ... + (-1)^n nCn is equal toThe sum of the binomial coefficients 10C0 + 10C1 + ... + 10C10 isEvaluate 3C0 + 3 x 3C1 + 9 x 3C2 + 27 x 3C3.Computing 102 cubed as (100 + 2)^3 givesComputing 101 to the fourth power as (100 + 1)^4 givesIn showing (1.01)^1000000 > 10,000, the term 1000000C1 x 0.01 equalsWhen 6^n - 5n is divided by 25, the remainder is alwaysFor every positive integer n, 9^(n+1) - 8n - 9 is divisible byFor distinct integers a and b, a^n - b^n always has which factor?Using the result for (a + b)^4 - (a - b)^4, (sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4 equalsThe value of (sqrt2 + 1)^6 + (sqrt2 - 1)^6 isUsing the first three terms of its expansion, (0.99)^5 is approximatelyChu-shi-kie of China wrote the coefficient triangle inWho first introduced the term binomial coefficients, around 1544?Oughtred listed binomial coefficients for n = 1 to 10 in which year?Bombelli (1572) gave the coefficients of (a + b)^n up to which n?Pingla wrote his book Chhanda shastra aroundPascal's treatise containing the binomial theorem was published posthumously inIn what year did the French thinker Blaise Pascal pass away?The standard proof of the binomial theorem for positive integers n usesIn the inductive step, kCr + kC(r-1) is replaced byFor any non-negative integer n, the value of nC0 isThe combination nCr is defined asHow many binomial coefficients appear in the expansion of (a + b)^20?In summation form, (a + b)^n is the sum over k ofWriting (x^2 + 3/x)^4 out, the leading term isIn (x^2 + 3/x)^4, the term in x^5 has multiplierThe final term of (x^2 + 3/x)^4 isWriting (1 - x)^6 out, the x term has multiplierUsing the first two terms of (1 + 0.1)^10000, the partial sum isThe binomial theorem for a finite expansion of (a + b)^n needs n to be aExpanded fully, (2x - 3)^6 begins with